Dirichlet series
| L(s) = 1 | + 2·5-s + 19·9-s + 14·11-s − 14·19-s − 5·25-s + 22·29-s + 20·31-s − 32·41-s + 38·45-s + 66·49-s + 28·55-s − 22·59-s + 10·61-s + 28·71-s − 64·79-s + 175·81-s + 48·89-s − 28·95-s + 266·99-s − 58·101-s + 54·109-s + 47·121-s − 24·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 19/3·9-s + 4.22·11-s − 3.21·19-s − 25-s + 4.08·29-s + 3.59·31-s − 4.99·41-s + 5.66·45-s + 66/7·49-s + 3.77·55-s − 2.86·59-s + 1.28·61-s + 3.32·71-s − 7.20·79-s + 19.4·81-s + 5.08·89-s − 2.87·95-s + 26.7·99-s − 5.77·101-s + 5.17·109-s + 4.27·121-s − 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 5^{14} \cdot 23^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 5^{14} \cdot 23^{14}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(28\) |
| Conductor: | \(2^{56} \cdot 5^{14} \cdot 23^{14}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.18438\times 10^{16}\) |
| Root analytic conductor: | \(3.83307\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((28,\ 2^{56} \cdot 5^{14} \cdot 23^{14} ,\ ( \ : [1/2]^{14} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(190.6705587\) |
| \(L(\frac12)\) | \(\approx\) | \(190.6705587\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 2 T + 9 T^{2} - 4 T^{3} + 7 T^{4} + 2 T^{5} + 127 T^{6} - 344 T^{7} + 127 p T^{8} + 2 p^{2} T^{9} + 7 p^{3} T^{10} - 4 p^{4} T^{11} + 9 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( ( 1 + T^{2} )^{7} \) | |
| good | 3 | \( 1 - 19 T^{2} + 62 p T^{4} - 1250 T^{6} + 2161 p T^{8} - 9244 p T^{10} + 101651 T^{12} - 325376 T^{14} + 101651 p^{2} T^{16} - 9244 p^{5} T^{18} + 2161 p^{7} T^{20} - 1250 p^{8} T^{22} + 62 p^{11} T^{24} - 19 p^{12} T^{26} + p^{14} T^{28} \) |
| 7 | \( 1 - 66 T^{2} + 2183 T^{4} - 47459 T^{6} + 753577 T^{8} - 9213346 T^{10} + 89256767 T^{12} - 695230850 T^{14} + 89256767 p^{2} T^{16} - 9213346 p^{4} T^{18} + 753577 p^{6} T^{20} - 47459 p^{8} T^{22} + 2183 p^{10} T^{24} - 66 p^{12} T^{26} + p^{14} T^{28} \) | |
| 11 | \( ( 1 - 7 T + 50 T^{2} - 24 p T^{3} + 1304 T^{4} - 5217 T^{5} + 20319 T^{6} - 69648 T^{7} + 20319 p T^{8} - 5217 p^{2} T^{9} + 1304 p^{3} T^{10} - 24 p^{5} T^{11} + 50 p^{5} T^{12} - 7 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 13 | \( 1 - 107 T^{2} + 5394 T^{4} - 172078 T^{6} + 305247 p T^{8} - 72258180 T^{10} + 1112944891 T^{12} - 15218250732 T^{14} + 1112944891 p^{2} T^{16} - 72258180 p^{4} T^{18} + 305247 p^{7} T^{20} - 172078 p^{8} T^{22} + 5394 p^{10} T^{24} - 107 p^{12} T^{26} + p^{14} T^{28} \) | |
| 17 | \( 1 - 122 T^{2} + 7003 T^{4} - 248683 T^{6} + 6055449 T^{8} - 107644814 T^{10} + 1549072811 T^{12} - 23227734218 T^{14} + 1549072811 p^{2} T^{16} - 107644814 p^{4} T^{18} + 6055449 p^{6} T^{20} - 248683 p^{8} T^{22} + 7003 p^{10} T^{24} - 122 p^{12} T^{26} + p^{14} T^{28} \) | |
| 19 | \( ( 1 + 7 T + 80 T^{2} + 560 T^{3} + 3576 T^{4} + 20921 T^{5} + 103617 T^{6} + 488640 T^{7} + 103617 p T^{8} + 20921 p^{2} T^{9} + 3576 p^{3} T^{10} + 560 p^{4} T^{11} + 80 p^{5} T^{12} + 7 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 29 | \( ( 1 - 11 T + 147 T^{2} - 984 T^{3} + 8570 T^{4} - 46128 T^{5} + 336468 T^{6} - 1566138 T^{7} + 336468 p T^{8} - 46128 p^{2} T^{9} + 8570 p^{3} T^{10} - 984 p^{4} T^{11} + 147 p^{5} T^{12} - 11 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 31 | \( ( 1 - 10 T + 107 T^{2} - 617 T^{3} + 5296 T^{4} - 28550 T^{5} + 208730 T^{6} - 954795 T^{7} + 208730 p T^{8} - 28550 p^{2} T^{9} + 5296 p^{3} T^{10} - 617 p^{4} T^{11} + 107 p^{5} T^{12} - 10 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 37 | \( 1 - 293 T^{2} + 44239 T^{4} - 4504978 T^{6} + 343871477 T^{8} - 20751361259 T^{10} + 1019473928875 T^{12} - 41393907437468 T^{14} + 1019473928875 p^{2} T^{16} - 20751361259 p^{4} T^{18} + 343871477 p^{6} T^{20} - 4504978 p^{8} T^{22} + 44239 p^{10} T^{24} - 293 p^{12} T^{26} + p^{14} T^{28} \) | |
| 41 | \( ( 1 + 16 T + 299 T^{2} + 2981 T^{3} + 33474 T^{4} + 254854 T^{5} + 2163504 T^{6} + 13191531 T^{7} + 2163504 p T^{8} + 254854 p^{2} T^{9} + 33474 p^{3} T^{10} + 2981 p^{4} T^{11} + 299 p^{5} T^{12} + 16 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 43 | \( 1 - 506 T^{2} + 120767 T^{4} - 18086140 T^{6} + 1906966741 T^{8} - 150435650374 T^{10} + 9194894688539 T^{12} - 443672445532488 T^{14} + 9194894688539 p^{2} T^{16} - 150435650374 p^{4} T^{18} + 1906966741 p^{6} T^{20} - 18086140 p^{8} T^{22} + 120767 p^{10} T^{24} - 506 p^{12} T^{26} + p^{14} T^{28} \) | |
| 47 | \( 1 - 266 T^{2} + 38569 T^{4} - 4017312 T^{6} + 329764176 T^{8} - 22494973712 T^{10} + 1310292484982 T^{12} - 66022395493804 T^{14} + 1310292484982 p^{2} T^{16} - 22494973712 p^{4} T^{18} + 329764176 p^{6} T^{20} - 4017312 p^{8} T^{22} + 38569 p^{10} T^{24} - 266 p^{12} T^{26} + p^{14} T^{28} \) | |
| 53 | \( 1 - 397 T^{2} + 82271 T^{4} - 11591450 T^{6} + 1231873957 T^{8} - 104206151155 T^{10} + 7233652549035 T^{12} - 418745375631148 T^{14} + 7233652549035 p^{2} T^{16} - 104206151155 p^{4} T^{18} + 1231873957 p^{6} T^{20} - 11591450 p^{8} T^{22} + 82271 p^{10} T^{24} - 397 p^{12} T^{26} + p^{14} T^{28} \) | |
| 59 | \( ( 1 + 11 T + 161 T^{2} + 1606 T^{3} + 15241 T^{4} + 104989 T^{5} + 902577 T^{6} + 6212660 T^{7} + 902577 p T^{8} + 104989 p^{2} T^{9} + 15241 p^{3} T^{10} + 1606 p^{4} T^{11} + 161 p^{5} T^{12} + 11 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 61 | \( ( 1 - 5 T + 112 T^{2} + 218 T^{3} + 7820 T^{4} + 11169 T^{5} + 779059 T^{6} + 139788 T^{7} + 779059 p T^{8} + 11169 p^{2} T^{9} + 7820 p^{3} T^{10} + 218 p^{4} T^{11} + 112 p^{5} T^{12} - 5 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 67 | \( 1 - 369 T^{2} + 65479 T^{4} - 7261706 T^{6} + 565267357 T^{8} - 34299289407 T^{10} + 1895368357035 T^{12} - 115358926660780 T^{14} + 1895368357035 p^{2} T^{16} - 34299289407 p^{4} T^{18} + 565267357 p^{6} T^{20} - 7261706 p^{8} T^{22} + 65479 p^{10} T^{24} - 369 p^{12} T^{26} + p^{14} T^{28} \) | |
| 71 | \( ( 1 - 14 T + 341 T^{2} - 3253 T^{3} + 50270 T^{4} - 415786 T^{5} + 5172636 T^{6} - 36898691 T^{7} + 5172636 p T^{8} - 415786 p^{2} T^{9} + 50270 p^{3} T^{10} - 3253 p^{4} T^{11} + 341 p^{5} T^{12} - 14 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 73 | \( 1 - 862 T^{2} + 354217 T^{4} - 91970752 T^{6} + 16854894880 T^{8} - 2306015009424 T^{10} + 242867435234150 T^{12} - 20001032688147428 T^{14} + 242867435234150 p^{2} T^{16} - 2306015009424 p^{4} T^{18} + 16854894880 p^{6} T^{20} - 91970752 p^{8} T^{22} + 354217 p^{10} T^{24} - 862 p^{12} T^{26} + p^{14} T^{28} \) | |
| 79 | \( ( 1 + 32 T + 719 T^{2} + 12292 T^{3} + 173907 T^{4} + 2116176 T^{5} + 22653621 T^{6} + 212954872 T^{7} + 22653621 p T^{8} + 2116176 p^{2} T^{9} + 173907 p^{3} T^{10} + 12292 p^{4} T^{11} + 719 p^{5} T^{12} + 32 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 83 | \( 1 - 717 T^{2} + 246543 T^{4} - 54114654 T^{6} + 8558105653 T^{8} - 1052493882963 T^{10} + 107035491394123 T^{12} - 9444801982758308 T^{14} + 107035491394123 p^{2} T^{16} - 1052493882963 p^{4} T^{18} + 8558105653 p^{6} T^{20} - 54114654 p^{8} T^{22} + 246543 p^{10} T^{24} - 717 p^{12} T^{26} + p^{14} T^{28} \) | |
| 89 | \( ( 1 - 24 T + 669 T^{2} - 10556 T^{3} + 175975 T^{4} - 2095656 T^{5} + 25712747 T^{6} - 239948584 T^{7} + 25712747 p T^{8} - 2095656 p^{2} T^{9} + 175975 p^{3} T^{10} - 10556 p^{4} T^{11} + 669 p^{5} T^{12} - 24 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 97 | \( 1 - 571 T^{2} + 181248 T^{4} - 40462120 T^{6} + 7078397138 T^{8} - 1020702744285 T^{10} + 124694580796541 T^{12} - 13046430227502176 T^{14} + 124694580796541 p^{2} T^{16} - 1020702744285 p^{4} T^{18} + 7078397138 p^{6} T^{20} - 40462120 p^{8} T^{22} + 181248 p^{10} T^{24} - 571 p^{12} T^{26} + p^{14} T^{28} \) | |
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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
−2.61078090241907958418957985112, −2.54472415112629570515935724888, −2.31372077556602765889268833238, −2.16656295679753781805609161575, −2.15837395442535619269949189105, −2.05696802685984572610530490511, −1.91321283659269739923842395608, −1.87801128986824782543354927746, −1.87329077680714895846456021922, −1.68947733212085388026553443830, −1.68726517466984282971835180240, −1.61313879119983150664295349983, −1.56732596191691130658639781153, −1.55928326413005026797748177561, −1.49131794945646159828252517923, −1.18036471951747857334829110815, −1.03639338196945115734507379872, −0.979921711663664490485275325013, −0.976671932290797102505285524924, −0.957199736531995539253762338435, −0.847875693579928602973065173965, −0.74721543601711695574083415855, −0.58066761152066023840967642971, −0.45735994679497252389361585157, −0.18855380373389922085233442517, 0.18855380373389922085233442517, 0.45735994679497252389361585157, 0.58066761152066023840967642971, 0.74721543601711695574083415855, 0.847875693579928602973065173965, 0.957199736531995539253762338435, 0.976671932290797102505285524924, 0.979921711663664490485275325013, 1.03639338196945115734507379872, 1.18036471951747857334829110815, 1.49131794945646159828252517923, 1.55928326413005026797748177561, 1.56732596191691130658639781153, 1.61313879119983150664295349983, 1.68726517466984282971835180240, 1.68947733212085388026553443830, 1.87329077680714895846456021922, 1.87801128986824782543354927746, 1.91321283659269739923842395608, 2.05696802685984572610530490511, 2.15837395442535619269949189105, 2.16656295679753781805609161575, 2.31372077556602765889268833238, 2.54472415112629570515935724888, 2.61078090241907958418957985112
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.