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Properties

Label 8-8034e4-1.1-c1e4-0-0
Degree $8$
Conductor $4.166\times 10^{15}$
Sign $1$
Analytic cond. $1.69369\times 10^{7}$
Root an. cond. $8.00948$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 4·3-s + 10·4-s + 2·5-s + 16·6-s + 5·7-s + 20·8-s + 10·9-s + 8·10-s + 9·11-s + 40·12-s + 4·13-s + 20·14-s + 8·15-s + 35·16-s + 8·17-s + 40·18-s − 6·19-s + 20·20-s + 20·21-s + 36·22-s + 6·23-s + 80·24-s − 5·25-s + 16·26-s + 20·27-s + 50·28-s + ⋯
L(s)  = 1  + 2.82·2-s + 2.30·3-s + 5·4-s + 0.894·5-s + 6.53·6-s + 1.88·7-s + 7.07·8-s + 10/3·9-s + 2.52·10-s + 2.71·11-s + 11.5·12-s + 1.10·13-s + 5.34·14-s + 2.06·15-s + 35/4·16-s + 1.94·17-s + 9.42·18-s − 1.37·19-s + 4.47·20-s + 4.36·21-s + 7.67·22-s + 1.25·23-s + 16.3·24-s − 25-s + 3.13·26-s + 3.84·27-s + 9.44·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 13^{4} \cdot 103^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 13^{4} \cdot 103^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 13^{4} \cdot 103^{4}\)
Sign: $1$
Analytic conductor: \(1.69369\times 10^{7}\)
Root analytic conductor: \(8.00948\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 13^{4} \cdot 103^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(764.8472754\)
\(L(\frac12)\) \(\approx\) \(764.8472754\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( ( 1 - T )^{4} \)
3$C_1$ \( ( 1 - T )^{4} \)
13$C_1$ \( ( 1 - T )^{4} \)
103$C_1$ \( ( 1 + T )^{4} \)
good5$C_2 \wr S_4$ \( 1 - 2 T + 9 T^{2} - 7 T^{3} + 44 T^{4} - 7 p T^{5} + 9 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr S_4$ \( 1 - 5 T + 17 T^{2} - 23 T^{3} + 53 T^{4} - 23 p T^{5} + 17 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 9 T + 54 T^{2} - 233 T^{3} + 874 T^{4} - 233 p T^{5} + 54 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
19$C_2 \wr S_4$ \( 1 + 6 T + 52 T^{2} + 222 T^{3} + 1382 T^{4} + 222 p T^{5} + 52 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 6 T + 68 T^{2} - 336 T^{3} + 2261 T^{4} - 336 p T^{5} + 68 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 89 T^{2} - 13 T^{3} + 3600 T^{4} - 13 p T^{5} + 89 p^{2} T^{6} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 6 T + 100 T^{2} - 438 T^{3} + 4406 T^{4} - 438 p T^{5} + 100 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 17 T + 236 T^{2} + 2003 T^{3} + 14654 T^{4} + 2003 p T^{5} + 236 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 16 T + 210 T^{2} - 1912 T^{3} + 13867 T^{4} - 1912 p T^{5} + 210 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 23 T + 350 T^{2} - 3511 T^{3} + 26970 T^{4} - 3511 p T^{5} + 350 p^{2} T^{6} - 23 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 + T + 103 T^{2} + 97 T^{3} + 6849 T^{4} + 97 p T^{5} + 103 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 18 T + 176 T^{2} - 22 p T^{3} + 8190 T^{4} - 22 p^{2} T^{5} + 176 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 20 T + 349 T^{2} - 3763 T^{3} + 34340 T^{4} - 3763 p T^{5} + 349 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 9 T + 222 T^{2} + 1439 T^{3} + 19410 T^{4} + 1439 p T^{5} + 222 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 8 T + 239 T^{2} - 1447 T^{3} + 348 p T^{4} - 1447 p T^{5} + 239 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 21 T + 207 T^{2} + 512 T^{3} - 1568 T^{4} + 512 p T^{5} + 207 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 22 T + 407 T^{2} - 4705 T^{3} + 48122 T^{4} - 4705 p T^{5} + 407 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 22 T + 404 T^{2} + 4950 T^{3} + 50198 T^{4} + 4950 p T^{5} + 404 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 18 T + 441 T^{2} + 4723 T^{3} + 59548 T^{4} + 4723 p T^{5} + 441 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 14 T + 324 T^{2} + 3042 T^{3} + 40518 T^{4} + 3042 p T^{5} + 324 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 20 T + 144 T^{2} + 1724 T^{3} - 32354 T^{4} + 1724 p T^{5} + 144 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.62230134184147739505637436708, −5.25302045338633984573134029181, −4.97732477468019507040656148136, −4.96302910589801869381739748147, −4.63532730923812091041570715798, −4.33681615770150313892221954657, −4.23681176740293500611206882139, −4.15562820388462328855168015175, −4.11372213195565635506253585987, −3.78707218135262712341301829948, −3.62806726909889854435948377805, −3.61981604917821591341934800357, −3.32185492852043244103119991020, −3.00775012750816932149640105608, −2.91241553381412929112174742802, −2.60658941365710428003284680568, −2.43564895340626286214817983712, −2.19001649541939807625351605937, −2.00091833446522303595393528195, −1.87323102091486971876036711957, −1.71592333224264253606442450300, −1.25903961686184811237576924425, −1.04954133379681511341564652904, −1.01472946209211290460240658181, −0.988330683834742524462839754422, 0.988330683834742524462839754422, 1.01472946209211290460240658181, 1.04954133379681511341564652904, 1.25903961686184811237576924425, 1.71592333224264253606442450300, 1.87323102091486971876036711957, 2.00091833446522303595393528195, 2.19001649541939807625351605937, 2.43564895340626286214817983712, 2.60658941365710428003284680568, 2.91241553381412929112174742802, 3.00775012750816932149640105608, 3.32185492852043244103119991020, 3.61981604917821591341934800357, 3.62806726909889854435948377805, 3.78707218135262712341301829948, 4.11372213195565635506253585987, 4.15562820388462328855168015175, 4.23681176740293500611206882139, 4.33681615770150313892221954657, 4.63532730923812091041570715798, 4.96302910589801869381739748147, 4.97732477468019507040656148136, 5.25302045338633984573134029181, 5.62230134184147739505637436708

Graph of the $Z$-function along the critical line