Dirichlet series
| L(s) = 1 | + 4·2-s + 10·4-s + 32·8-s + 10·9-s − 4·11-s + 87·16-s + 40·18-s − 16·22-s + 12·23-s − 52·25-s + 8·29-s + 196·32-s + 100·36-s − 8·37-s + 32·43-s − 40·44-s + 48·46-s − 208·50-s − 8·53-s + 32·58-s + 438·64-s + 20·67-s + 8·71-s + 320·72-s − 32·74-s − 8·79-s + 67·81-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 5·4-s + 11.3·8-s + 10/3·9-s − 1.20·11-s + 87/4·16-s + 9.42·18-s − 3.41·22-s + 2.50·23-s − 10.3·25-s + 1.48·29-s + 34.6·32-s + 50/3·36-s − 1.31·37-s + 4.87·43-s − 6.03·44-s + 7.07·46-s − 29.4·50-s − 1.09·53-s + 4.20·58-s + 54.7·64-s + 2.44·67-s + 0.949·71-s + 37.7·72-s − 3.71·74-s − 0.900·79-s + 67/9·81-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{32} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{32} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(7^{32} \cdot 13^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.00754\times 10^{11}\) |
| Root analytic conductor: | \(2.25532\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 7^{32} \cdot 13^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(694.6913152\) |
| \(L(\frac12)\) | \(\approx\) | \(694.6913152\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 + 34 T^{2} + 430 T^{4} + 4984 T^{6} + 76567 T^{8} + 4984 p^{2} T^{10} + 430 p^{4} T^{12} + 34 p^{6} T^{14} + p^{8} T^{16} \) | |
| good | 2 | \( ( 1 - p T + T^{2} - 3 p T^{3} + 5 p T^{4} - p^{2} T^{5} + 27 T^{6} - 5 p^{3} T^{7} + 13 T^{8} - 5 p^{4} T^{9} + 27 p^{2} T^{10} - p^{5} T^{11} + 5 p^{5} T^{12} - 3 p^{6} T^{13} + p^{6} T^{14} - p^{8} T^{15} + p^{8} T^{16} )^{2} \) |
| 3 | \( 1 - 10 T^{2} + 11 p T^{4} - 58 T^{6} + 124 p T^{8} - 2116 T^{10} + 6131 T^{12} - 1774 p^{2} T^{14} + 50155 T^{16} - 1774 p^{4} T^{18} + 6131 p^{4} T^{20} - 2116 p^{6} T^{22} + 124 p^{9} T^{24} - 58 p^{10} T^{26} + 11 p^{13} T^{28} - 10 p^{14} T^{30} + p^{16} T^{32} \) | |
| 5 | \( ( 1 + 26 T^{2} + 311 T^{4} + 2358 T^{6} + 13281 T^{8} + 2358 p^{2} T^{10} + 311 p^{4} T^{12} + 26 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 11 | \( ( 1 + 2 T - 35 T^{2} - 30 T^{3} + 802 T^{4} + 256 T^{5} - 12663 T^{6} - 1238 T^{7} + 153211 T^{8} - 1238 p T^{9} - 12663 p^{2} T^{10} + 256 p^{3} T^{11} + 802 p^{4} T^{12} - 30 p^{5} T^{13} - 35 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 17 | \( 1 - 78 T^{2} + 2942 T^{4} - 74036 T^{6} + 1506893 T^{8} - 29044204 T^{10} + 585942562 T^{12} - 11972588178 T^{14} + 220591800628 T^{16} - 11972588178 p^{2} T^{18} + 585942562 p^{4} T^{20} - 29044204 p^{6} T^{22} + 1506893 p^{8} T^{24} - 74036 p^{10} T^{26} + 2942 p^{12} T^{28} - 78 p^{14} T^{30} + p^{16} T^{32} \) | |
| 19 | \( 1 - 58 T^{2} + 1661 T^{4} - 9818 T^{6} - 676340 T^{8} + 23829108 T^{10} - 206707529 T^{12} - 5249133354 T^{14} + 201079481139 T^{16} - 5249133354 p^{2} T^{18} - 206707529 p^{4} T^{20} + 23829108 p^{6} T^{22} - 676340 p^{8} T^{24} - 9818 p^{10} T^{26} + 1661 p^{12} T^{28} - 58 p^{14} T^{30} + p^{16} T^{32} \) | |
| 23 | \( ( 1 - 6 T - 42 T^{2} + 248 T^{3} + 1306 T^{4} - 4598 T^{5} - 43856 T^{6} + 22390 T^{7} + 1346875 T^{8} + 22390 p T^{9} - 43856 p^{2} T^{10} - 4598 p^{3} T^{11} + 1306 p^{4} T^{12} + 248 p^{5} T^{13} - 42 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 29 | \( ( 1 - 4 T - 15 T^{2} + 192 T^{3} - 1568 T^{4} + 5696 T^{5} - p T^{6} - 217482 T^{7} + 2139619 T^{8} - 217482 p T^{9} - p^{3} T^{10} + 5696 p^{3} T^{11} - 1568 p^{4} T^{12} + 192 p^{5} T^{13} - 15 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 31 | \( ( 1 + 178 T^{2} + 15406 T^{4} + 835752 T^{6} + 30991415 T^{8} + 835752 p^{2} T^{10} + 15406 p^{4} T^{12} + 178 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 37 | \( ( 1 + 4 T - 46 T^{2} + 152 T^{3} + 2182 T^{4} - 7008 T^{5} + 73816 T^{6} + 392460 T^{7} - 3440049 T^{8} + 392460 p T^{9} + 73816 p^{2} T^{10} - 7008 p^{3} T^{11} + 2182 p^{4} T^{12} + 152 p^{5} T^{13} - 46 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 41 | \( 1 - 152 T^{2} + 16324 T^{4} - 1214800 T^{6} + 75014282 T^{8} - 3793839848 T^{10} + 172337071760 T^{12} - 7119768845032 T^{14} + 293658219524371 T^{16} - 7119768845032 p^{2} T^{18} + 172337071760 p^{4} T^{20} - 3793839848 p^{6} T^{22} + 75014282 p^{8} T^{24} - 1214800 p^{10} T^{26} + 16324 p^{12} T^{28} - 152 p^{14} T^{30} + p^{16} T^{32} \) | |
| 43 | \( ( 1 - 16 T + 99 T^{2} - 472 T^{3} + 1730 T^{4} + 7168 T^{5} - 116817 T^{6} + 1124260 T^{7} - 9477617 T^{8} + 1124260 p T^{9} - 116817 p^{2} T^{10} + 7168 p^{3} T^{11} + 1730 p^{4} T^{12} - 472 p^{5} T^{13} + 99 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 47 | \( ( 1 + 150 T^{2} + 11494 T^{4} + 707936 T^{6} + 37180919 T^{8} + 707936 p^{2} T^{10} + 11494 p^{4} T^{12} + 150 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 53 | \( ( 1 + 2 T + 132 T^{2} + 248 T^{3} + 8645 T^{4} + 248 p T^{5} + 132 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 59 | \( 1 - 278 T^{2} + 40998 T^{4} - 3814004 T^{6} + 237186965 T^{8} - 9386021884 T^{10} + 154854604410 T^{12} + 7674718497878 T^{14} - 788274714787388 T^{16} + 7674718497878 p^{2} T^{18} + 154854604410 p^{4} T^{20} - 9386021884 p^{6} T^{22} + 237186965 p^{8} T^{24} - 3814004 p^{10} T^{26} + 40998 p^{12} T^{28} - 278 p^{14} T^{30} + p^{16} T^{32} \) | |
| 61 | \( 1 - 380 T^{2} + 76544 T^{4} - 10897784 T^{6} + 1227826114 T^{8} - 116331500276 T^{10} + 9578956327264 T^{12} - 696692904086100 T^{14} + 45045833784892083 T^{16} - 696692904086100 p^{2} T^{18} + 9578956327264 p^{4} T^{20} - 116331500276 p^{6} T^{22} + 1227826114 p^{8} T^{24} - 10897784 p^{10} T^{26} + 76544 p^{12} T^{28} - 380 p^{14} T^{30} + p^{16} T^{32} \) | |
| 67 | \( ( 1 - 10 T - 162 T^{2} + 956 T^{3} + 24917 T^{4} - 71312 T^{5} - 2420694 T^{6} + 1789858 T^{7} + 185730724 T^{8} + 1789858 p T^{9} - 2420694 p^{2} T^{10} - 71312 p^{3} T^{11} + 24917 p^{4} T^{12} + 956 p^{5} T^{13} - 162 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 71 | \( ( 1 - 4 T - 102 T^{2} + 696 T^{3} - 1042 T^{4} - 9336 T^{5} - 125464 T^{6} - 1401412 T^{7} + 51152511 T^{8} - 1401412 p T^{9} - 125464 p^{2} T^{10} - 9336 p^{3} T^{11} - 1042 p^{4} T^{12} + 696 p^{5} T^{13} - 102 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 73 | \( ( 1 + 156 T^{2} + 7232 T^{4} + 2020 p T^{6} + 11253390 T^{8} + 2020 p^{3} T^{10} + 7232 p^{4} T^{12} + 156 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 79 | \( ( 1 + 2 T + 38 T^{2} - 238 T^{3} + 1686 T^{4} - 238 p T^{5} + 38 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{4} \) | |
| 83 | \( ( 1 + 314 T^{2} + 46430 T^{4} + 4837464 T^{6} + 427877847 T^{8} + 4837464 p^{2} T^{10} + 46430 p^{4} T^{12} + 314 p^{6} T^{14} + p^{8} T^{16} )^{2} \) | |
| 89 | \( 1 - 106 T^{2} - 6635 T^{4} + 2260150 T^{6} - 56645140 T^{8} - 20540045260 T^{10} + 2028500040815 T^{12} + 92860376641870 T^{14} - 22701882367538285 T^{16} + 92860376641870 p^{2} T^{18} + 2028500040815 p^{4} T^{20} - 20540045260 p^{6} T^{22} - 56645140 p^{8} T^{24} + 2260150 p^{10} T^{26} - 6635 p^{12} T^{28} - 106 p^{14} T^{30} + p^{16} T^{32} \) | |
| 97 | \( 1 - 412 T^{2} + 87509 T^{4} - 11349760 T^{6} + 907888036 T^{8} - 32651756224 T^{10} - 1974791695085 T^{12} + 422191622326362 T^{14} - 45739383615060693 T^{16} + 422191622326362 p^{2} T^{18} - 1974791695085 p^{4} T^{20} - 32651756224 p^{6} T^{22} + 907888036 p^{8} T^{24} - 11349760 p^{10} T^{26} + 87509 p^{12} T^{28} - 412 p^{14} T^{30} + p^{16} T^{32} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.82971545980281424348696263257, −2.71465563260493320135431246918, −2.53676864066042077082901779177, −2.50572230772502125393385968961, −2.50453516339212674287377002269, −2.40507548721078497229187371039, −2.35079628947152365897669539918, −2.34584168565053353127241613396, −2.11912395278826651169602815346, −2.01061848449569760489930310482, −1.94674195769371218729679271588, −1.87272041631664334366659867596, −1.80515769542522829470395819796, −1.66158221927596276142838311504, −1.59058216955735166432475461521, −1.47572498512007748167774136262, −1.46984545366465835967030062283, −1.42568431320038204824636494391, −1.40864519121509738458608423867, −1.39684944977077239307035708651, −0.71053457165757075059635209742, −0.68533658273075388308636848416, −0.68030151007276984111322096984, −0.57490128015347619171821733892, −0.57451372504531519107677728228, 0.57451372504531519107677728228, 0.57490128015347619171821733892, 0.68030151007276984111322096984, 0.68533658273075388308636848416, 0.71053457165757075059635209742, 1.39684944977077239307035708651, 1.40864519121509738458608423867, 1.42568431320038204824636494391, 1.46984545366465835967030062283, 1.47572498512007748167774136262, 1.59058216955735166432475461521, 1.66158221927596276142838311504, 1.80515769542522829470395819796, 1.87272041631664334366659867596, 1.94674195769371218729679271588, 2.01061848449569760489930310482, 2.11912395278826651169602815346, 2.34584168565053353127241613396, 2.35079628947152365897669539918, 2.40507548721078497229187371039, 2.50453516339212674287377002269, 2.50572230772502125393385968961, 2.53676864066042077082901779177, 2.71465563260493320135431246918, 2.82971545980281424348696263257
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.