Dirichlet series
| L(s) = 1 | + 3·3-s − 2·5-s + 2·7-s + 13·9-s + 12·11-s + 4·13-s − 6·15-s − 7·17-s + 3·19-s + 6·21-s + 18·23-s + 12·25-s + 24·27-s + 10·29-s + 10·31-s + 36·33-s − 4·35-s − 11·37-s + 12·39-s − 30·43-s − 26·45-s − 15·47-s + 12·49-s − 21·51-s − 8·53-s − 24·55-s + 9·57-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.894·5-s + 0.755·7-s + 13/3·9-s + 3.61·11-s + 1.10·13-s − 1.54·15-s − 1.69·17-s + 0.688·19-s + 1.30·21-s + 3.75·23-s + 12/5·25-s + 4.61·27-s + 1.85·29-s + 1.79·31-s + 6.26·33-s − 0.676·35-s − 1.80·37-s + 1.92·39-s − 4.57·43-s − 3.87·45-s − 2.18·47-s + 12/7·49-s − 2.94·51-s − 1.09·53-s − 3.23·55-s + 1.19·57-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{42} \cdot 7^{14} \cdot 17^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{42} \cdot 7^{14} \cdot 17^{14}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(28\) |
| Conductor: | \(2^{42} \cdot 7^{14} \cdot 17^{14}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.15177\times 10^{12}\) |
| Root analytic conductor: | \(2.75712\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((28,\ 2^{42} \cdot 7^{14} \cdot 17^{14} ,\ ( \ : [1/2]^{14} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(25.57176315\) |
| \(L(\frac12)\) | \(\approx\) | \(25.57176315\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 2 T - 8 T^{2} + 27 T^{3} - 66 T^{4} + 30 T^{5} + 523 T^{6} - 1682 T^{7} + 523 p T^{8} + 30 p^{2} T^{9} - 66 p^{3} T^{10} + 27 p^{4} T^{11} - 8 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( ( 1 + T + T^{2} )^{7} \) | |
| good | 3 | \( 1 - p T - 4 T^{2} + p^{3} T^{3} - 8 p T^{4} - 59 T^{5} + 43 p T^{6} - 37 T^{7} - 40 T^{8} + 41 T^{9} - 421 T^{10} + 85 p T^{11} + 433 p T^{12} + 278 T^{13} - 5804 T^{14} + 278 p T^{15} + 433 p^{3} T^{16} + 85 p^{4} T^{17} - 421 p^{4} T^{18} + 41 p^{5} T^{19} - 40 p^{6} T^{20} - 37 p^{7} T^{21} + 43 p^{9} T^{22} - 59 p^{9} T^{23} - 8 p^{11} T^{24} + p^{14} T^{25} - 4 p^{12} T^{26} - p^{14} T^{27} + p^{14} T^{28} \) |
| 5 | \( 1 + 2 T - 8 T^{2} + 14 T^{3} + 84 T^{4} - 138 T^{5} + 183 T^{6} + 1266 T^{7} - 2779 T^{8} + 3624 T^{9} + 18814 T^{10} - 41188 T^{11} + 43557 T^{12} + 172442 T^{13} - 428451 T^{14} + 172442 p T^{15} + 43557 p^{2} T^{16} - 41188 p^{3} T^{17} + 18814 p^{4} T^{18} + 3624 p^{5} T^{19} - 2779 p^{6} T^{20} + 1266 p^{7} T^{21} + 183 p^{8} T^{22} - 138 p^{9} T^{23} + 84 p^{10} T^{24} + 14 p^{11} T^{25} - 8 p^{12} T^{26} + 2 p^{13} T^{27} + p^{14} T^{28} \) | |
| 11 | \( 1 - 12 T + 26 T^{2} + 186 T^{3} - 3 p^{2} T^{4} - 402 p T^{5} + 9161 T^{6} + 3702 p T^{7} + 38356 T^{8} - 689484 T^{9} - 1318307 T^{10} + 3880962 T^{11} + 40512782 T^{12} - 38612916 T^{13} - 418124604 T^{14} - 38612916 p T^{15} + 40512782 p^{2} T^{16} + 3880962 p^{3} T^{17} - 1318307 p^{4} T^{18} - 689484 p^{5} T^{19} + 38356 p^{6} T^{20} + 3702 p^{8} T^{21} + 9161 p^{8} T^{22} - 402 p^{10} T^{23} - 3 p^{12} T^{24} + 186 p^{11} T^{25} + 26 p^{12} T^{26} - 12 p^{13} T^{27} + p^{14} T^{28} \) | |
| 13 | \( ( 1 - 2 T + 22 T^{2} - 115 T^{3} + 383 T^{4} - 1594 T^{5} + 7707 T^{6} - 16485 T^{7} + 7707 p T^{8} - 1594 p^{2} T^{9} + 383 p^{3} T^{10} - 115 p^{4} T^{11} + 22 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 19 | \( 1 - 3 T - 43 T^{2} + 162 T^{3} + 217 T^{4} - 2001 T^{5} + 4540 T^{6} + 4095 T^{7} + 290204 T^{8} - 866865 T^{9} - 8422196 T^{10} + 18189999 T^{11} + 34095894 T^{12} - 59851935 T^{13} + 805517454 T^{14} - 59851935 p T^{15} + 34095894 p^{2} T^{16} + 18189999 p^{3} T^{17} - 8422196 p^{4} T^{18} - 866865 p^{5} T^{19} + 290204 p^{6} T^{20} + 4095 p^{7} T^{21} + 4540 p^{8} T^{22} - 2001 p^{9} T^{23} + 217 p^{10} T^{24} + 162 p^{11} T^{25} - 43 p^{12} T^{26} - 3 p^{13} T^{27} + p^{14} T^{28} \) | |
| 23 | \( 1 - 18 T + 99 T^{2} - 26 T^{3} - 1184 T^{4} + 7250 T^{5} - 102067 T^{6} + 621110 T^{7} - 169007 T^{8} - 9328840 T^{9} + 19339526 T^{10} - 105353520 T^{11} + 1193772321 T^{12} + 458608474 T^{13} - 33619445603 T^{14} + 458608474 p T^{15} + 1193772321 p^{2} T^{16} - 105353520 p^{3} T^{17} + 19339526 p^{4} T^{18} - 9328840 p^{5} T^{19} - 169007 p^{6} T^{20} + 621110 p^{7} T^{21} - 102067 p^{8} T^{22} + 7250 p^{9} T^{23} - 1184 p^{10} T^{24} - 26 p^{11} T^{25} + 99 p^{12} T^{26} - 18 p^{13} T^{27} + p^{14} T^{28} \) | |
| 29 | \( ( 1 - 5 T + 102 T^{2} - 353 T^{3} + 3780 T^{4} - 8367 T^{5} + 69425 T^{6} - 121062 T^{7} + 69425 p T^{8} - 8367 p^{2} T^{9} + 3780 p^{3} T^{10} - 353 p^{4} T^{11} + 102 p^{5} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 31 | \( 1 - 10 T - 26 T^{2} + 426 T^{3} + 1118 T^{4} - 11484 T^{5} - 13229 T^{6} - 180692 T^{7} + 620831 T^{8} + 14889584 T^{9} + 1416478 p T^{10} - 724431984 T^{11} - 1710612385 T^{12} + 7448831424 T^{13} + 86605940661 T^{14} + 7448831424 p T^{15} - 1710612385 p^{2} T^{16} - 724431984 p^{3} T^{17} + 1416478 p^{5} T^{18} + 14889584 p^{5} T^{19} + 620831 p^{6} T^{20} - 180692 p^{7} T^{21} - 13229 p^{8} T^{22} - 11484 p^{9} T^{23} + 1118 p^{10} T^{24} + 426 p^{11} T^{25} - 26 p^{12} T^{26} - 10 p^{13} T^{27} + p^{14} T^{28} \) | |
| 37 | \( 1 + 11 T - 32 T^{2} - 731 T^{3} + 145 T^{4} + 39166 T^{5} + 89377 T^{6} - 1954050 T^{7} - 9436012 T^{8} + 66712966 T^{9} + 519958102 T^{10} - 1719295074 T^{11} - 24638230818 T^{12} + 20736767152 T^{13} + 979929919607 T^{14} + 20736767152 p T^{15} - 24638230818 p^{2} T^{16} - 1719295074 p^{3} T^{17} + 519958102 p^{4} T^{18} + 66712966 p^{5} T^{19} - 9436012 p^{6} T^{20} - 1954050 p^{7} T^{21} + 89377 p^{8} T^{22} + 39166 p^{9} T^{23} + 145 p^{10} T^{24} - 731 p^{11} T^{25} - 32 p^{12} T^{26} + 11 p^{13} T^{27} + p^{14} T^{28} \) | |
| 41 | \( ( 1 + 126 T^{2} + 261 T^{3} + 10356 T^{4} + 19860 T^{5} + 589781 T^{6} + 1117278 T^{7} + 589781 p T^{8} + 19860 p^{2} T^{9} + 10356 p^{3} T^{10} + 261 p^{4} T^{11} + 126 p^{5} T^{12} + p^{7} T^{14} )^{2} \) | |
| 43 | \( ( 1 + 15 T + 228 T^{2} + 2231 T^{3} + 21390 T^{4} + 160269 T^{5} + 1211599 T^{6} + 7845354 T^{7} + 1211599 p T^{8} + 160269 p^{2} T^{9} + 21390 p^{3} T^{10} + 2231 p^{4} T^{11} + 228 p^{5} T^{12} + 15 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 47 | \( 1 + 15 T + 5 T^{2} - 1098 T^{3} - 8291 T^{4} - 31947 T^{5} - 79356 T^{6} + 1195305 T^{7} + 25398576 T^{8} + 182985837 T^{9} + 572199708 T^{10} + 1589000241 T^{11} - 7120691294 T^{12} - 354608979045 T^{13} - 3641289181930 T^{14} - 354608979045 p T^{15} - 7120691294 p^{2} T^{16} + 1589000241 p^{3} T^{17} + 572199708 p^{4} T^{18} + 182985837 p^{5} T^{19} + 25398576 p^{6} T^{20} + 1195305 p^{7} T^{21} - 79356 p^{8} T^{22} - 31947 p^{9} T^{23} - 8291 p^{10} T^{24} - 1098 p^{11} T^{25} + 5 p^{12} T^{26} + 15 p^{13} T^{27} + p^{14} T^{28} \) | |
| 53 | \( 1 + 8 T - 126 T^{2} - 1222 T^{3} + 8633 T^{4} + 109856 T^{5} - 240123 T^{6} - 6152646 T^{7} - 10884682 T^{8} + 262381626 T^{9} + 2277215309 T^{10} - 7453625336 T^{11} - 187745155340 T^{12} + 131935341938 T^{13} + 11488138562352 T^{14} + 131935341938 p T^{15} - 187745155340 p^{2} T^{16} - 7453625336 p^{3} T^{17} + 2277215309 p^{4} T^{18} + 262381626 p^{5} T^{19} - 10884682 p^{6} T^{20} - 6152646 p^{7} T^{21} - 240123 p^{8} T^{22} + 109856 p^{9} T^{23} + 8633 p^{10} T^{24} - 1222 p^{11} T^{25} - 126 p^{12} T^{26} + 8 p^{13} T^{27} + p^{14} T^{28} \) | |
| 59 | \( 1 + 9 T - 262 T^{2} - 2207 T^{3} + 39975 T^{4} + 287534 T^{5} - 4620525 T^{6} - 25464390 T^{7} + 445239150 T^{8} + 1649310906 T^{9} - 36977951606 T^{10} - 75441311322 T^{11} + 2676117347340 T^{12} + 1665753668044 T^{13} - 169095715937095 T^{14} + 1665753668044 p T^{15} + 2676117347340 p^{2} T^{16} - 75441311322 p^{3} T^{17} - 36977951606 p^{4} T^{18} + 1649310906 p^{5} T^{19} + 445239150 p^{6} T^{20} - 25464390 p^{7} T^{21} - 4620525 p^{8} T^{22} + 287534 p^{9} T^{23} + 39975 p^{10} T^{24} - 2207 p^{11} T^{25} - 262 p^{12} T^{26} + 9 p^{13} T^{27} + p^{14} T^{28} \) | |
| 61 | \( 1 - 4 T - 196 T^{2} + 1750 T^{3} + 15950 T^{4} - 241620 T^{5} - 95293 T^{6} + 16297400 T^{7} - 76610189 T^{8} - 404575334 T^{9} + 5983325582 T^{10} - 13562982162 T^{11} - 174708737145 T^{12} + 772814871560 T^{13} + 2703279820449 T^{14} + 772814871560 p T^{15} - 174708737145 p^{2} T^{16} - 13562982162 p^{3} T^{17} + 5983325582 p^{4} T^{18} - 404575334 p^{5} T^{19} - 76610189 p^{6} T^{20} + 16297400 p^{7} T^{21} - 95293 p^{8} T^{22} - 241620 p^{9} T^{23} + 15950 p^{10} T^{24} + 1750 p^{11} T^{25} - 196 p^{12} T^{26} - 4 p^{13} T^{27} + p^{14} T^{28} \) | |
| 67 | \( 1 - 30 T + 310 T^{2} - 1706 T^{3} + 17430 T^{4} - 129908 T^{5} - 127117 T^{6} - 4144368 T^{7} + 119222915 T^{8} - 563294328 T^{9} + 6633656678 T^{10} - 98286785196 T^{11} + 445985436223 T^{12} - 3339924418972 T^{13} + 48092931622613 T^{14} - 3339924418972 p T^{15} + 445985436223 p^{2} T^{16} - 98286785196 p^{3} T^{17} + 6633656678 p^{4} T^{18} - 563294328 p^{5} T^{19} + 119222915 p^{6} T^{20} - 4144368 p^{7} T^{21} - 127117 p^{8} T^{22} - 129908 p^{9} T^{23} + 17430 p^{10} T^{24} - 1706 p^{11} T^{25} + 310 p^{12} T^{26} - 30 p^{13} T^{27} + p^{14} T^{28} \) | |
| 71 | \( ( 1 + 27 T + 686 T^{2} + 11085 T^{3} + 165905 T^{4} + 1913700 T^{5} + 20556189 T^{6} + 179316035 T^{7} + 20556189 p T^{8} + 1913700 p^{2} T^{9} + 165905 p^{3} T^{10} + 11085 p^{4} T^{11} + 686 p^{5} T^{12} + 27 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 73 | \( 1 - 5 T - 174 T^{2} - 661 T^{3} + 21285 T^{4} + 181754 T^{5} - 506555 T^{6} - 22036658 T^{7} - 98122020 T^{8} + 974191536 T^{9} + 12382103082 T^{10} + 13792505580 T^{11} - 642236086206 T^{12} - 1252331892800 T^{13} + 19357201563019 T^{14} - 1252331892800 p T^{15} - 642236086206 p^{2} T^{16} + 13792505580 p^{3} T^{17} + 12382103082 p^{4} T^{18} + 974191536 p^{5} T^{19} - 98122020 p^{6} T^{20} - 22036658 p^{7} T^{21} - 506555 p^{8} T^{22} + 181754 p^{9} T^{23} + 21285 p^{10} T^{24} - 661 p^{11} T^{25} - 174 p^{12} T^{26} - 5 p^{13} T^{27} + p^{14} T^{28} \) | |
| 79 | \( 1 + 3 T - 168 T^{2} - 1579 T^{3} + 5602 T^{4} + 174021 T^{5} + 658293 T^{6} - 640153 T^{7} - 41713224 T^{8} - 100397179 T^{9} - 1658813781 T^{10} - 79686095253 T^{11} - 662770784455 T^{12} + 4591402130186 T^{13} + 124122890872500 T^{14} + 4591402130186 p T^{15} - 662770784455 p^{2} T^{16} - 79686095253 p^{3} T^{17} - 1658813781 p^{4} T^{18} - 100397179 p^{5} T^{19} - 41713224 p^{6} T^{20} - 640153 p^{7} T^{21} + 658293 p^{8} T^{22} + 174021 p^{9} T^{23} + 5602 p^{10} T^{24} - 1579 p^{11} T^{25} - 168 p^{12} T^{26} + 3 p^{13} T^{27} + p^{14} T^{28} \) | |
| 83 | \( ( 1 + 12 T + 480 T^{2} + 4457 T^{3} + 101546 T^{4} + 759728 T^{5} + 12787335 T^{6} + 78372422 T^{7} + 12787335 p T^{8} + 759728 p^{2} T^{9} + 101546 p^{3} T^{10} + 4457 p^{4} T^{11} + 480 p^{5} T^{12} + 12 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 89 | \( 1 + 32 T + 55 T^{2} - 5784 T^{3} + 29753 T^{4} + 1576408 T^{5} - 4385012 T^{6} - 187610840 T^{7} + 1942709404 T^{8} + 26614534712 T^{9} - 271038120036 T^{10} - 1604405076088 T^{11} + 41728397377534 T^{12} + 94968306287320 T^{13} - 3661473460700950 T^{14} + 94968306287320 p T^{15} + 41728397377534 p^{2} T^{16} - 1604405076088 p^{3} T^{17} - 271038120036 p^{4} T^{18} + 26614534712 p^{5} T^{19} + 1942709404 p^{6} T^{20} - 187610840 p^{7} T^{21} - 4385012 p^{8} T^{22} + 1576408 p^{9} T^{23} + 29753 p^{10} T^{24} - 5784 p^{11} T^{25} + 55 p^{12} T^{26} + 32 p^{13} T^{27} + p^{14} T^{28} \) | |
| 97 | \( ( 1 + 39 T + 1183 T^{2} + 25359 T^{3} + 446417 T^{4} + 6530253 T^{5} + 80626447 T^{6} + 860866202 T^{7} + 80626447 p T^{8} + 6530253 p^{2} T^{9} + 446417 p^{3} T^{10} + 25359 p^{4} T^{11} + 1183 p^{5} T^{12} + 39 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{28} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.99819938010033219744608110285, −2.89390851814224971344200523980, −2.79493948995401755870986260349, −2.61212280635134108114879754614, −2.43096417009041680762747102315, −2.42519992073411251153432705831, −2.38271668934734490044901205159, −2.22663089284798662374783119198, −2.15257089470183560773633385673, −2.15051110752604431163936506574, −1.90673902612509504153804157929, −1.69467860787918211508752099091, −1.62686289489694471023092539400, −1.60035245367275614313060986139, −1.50364540969829796782009622685, −1.42097641865703575531690823604, −1.38210723833413904274964766367, −1.24661393091532433047803683104, −1.23479141827378629287156727331, −1.13776995009392037780198469228, −1.10755793296175349655073108109, −0.74072591600048106264929919760, −0.64305961211910949272491936265, −0.47523556361617211041202899459, −0.13517655868931313419489728677, 0.13517655868931313419489728677, 0.47523556361617211041202899459, 0.64305961211910949272491936265, 0.74072591600048106264929919760, 1.10755793296175349655073108109, 1.13776995009392037780198469228, 1.23479141827378629287156727331, 1.24661393091532433047803683104, 1.38210723833413904274964766367, 1.42097641865703575531690823604, 1.50364540969829796782009622685, 1.60035245367275614313060986139, 1.62686289489694471023092539400, 1.69467860787918211508752099091, 1.90673902612509504153804157929, 2.15051110752604431163936506574, 2.15257089470183560773633385673, 2.22663089284798662374783119198, 2.38271668934734490044901205159, 2.42519992073411251153432705831, 2.43096417009041680762747102315, 2.61212280635134108114879754614, 2.79493948995401755870986260349, 2.89390851814224971344200523980, 2.99819938010033219744608110285
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.