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Properties

Label 8-2541e4-1.1-c1e4-0-3
Degree $8$
Conductor $4.169\times 10^{13}$
Sign $1$
Analytic cond. $169483.$
Root an. cond. $4.50444$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4·3-s − 2·4-s − 4·5-s − 4·6-s − 4·7-s − 6·8-s + 10·9-s − 4·10-s + 8·12-s + 6·13-s − 4·14-s + 16·15-s − 4·16-s + 8·17-s + 10·18-s − 10·19-s + 8·20-s + 16·21-s − 10·23-s + 24·24-s + 4·25-s + 6·26-s − 20·27-s + 8·28-s + 16·30-s − 18·31-s + ⋯
L(s)  = 1  + 0.707·2-s − 2.30·3-s − 4-s − 1.78·5-s − 1.63·6-s − 1.51·7-s − 2.12·8-s + 10/3·9-s − 1.26·10-s + 2.30·12-s + 1.66·13-s − 1.06·14-s + 4.13·15-s − 16-s + 1.94·17-s + 2.35·18-s − 2.29·19-s + 1.78·20-s + 3.49·21-s − 2.08·23-s + 4.89·24-s + 4/5·25-s + 1.17·26-s − 3.84·27-s + 1.51·28-s + 2.92·30-s − 3.23·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 7^{4} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(169483.\)
Root analytic conductor: \(4.50444\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 3^{4} \cdot 7^{4} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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)$
bad3$C_1$ \( ( 1 + T )^{4} \)
7$C_1$ \( ( 1 + T )^{4} \)
11 \( 1 \)
good2$C_2 \wr C_2\wr C_2$ \( 1 - T + 3 T^{2} + T^{3} + 3 T^{4} + p T^{5} + 3 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
5$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 12 T^{2} + 36 T^{3} + 86 T^{4} + 36 p T^{5} + 12 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 44 T^{2} - 218 T^{3} + 822 T^{4} - 218 p T^{5} + 44 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 48 T^{2} - 264 T^{3} + 1358 T^{4} - 264 p T^{5} + 48 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 100 T^{2} + 30 p T^{3} + 3062 T^{4} + 30 p^{2} T^{5} + 100 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 83 T^{2} + 500 T^{3} + 2869 T^{4} + 500 p T^{5} + 83 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2^2 \wr C_2$ \( 1 + 37 T^{2} + 17 p T^{4} + 37 p^{2} T^{6} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 18 T + 192 T^{2} + 1362 T^{3} + 8238 T^{4} + 1362 p T^{5} + 192 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 71 T^{2} + 4 T^{3} + 2797 T^{4} + 4 p T^{5} + 71 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 60 T^{2} + 290 T^{3} - 2938 T^{4} + 290 p T^{5} + 60 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 69 T^{2} + 392 T^{3} + 4097 T^{4} + 392 p T^{5} + 69 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 108 T^{2} + 4 T^{3} + 4758 T^{4} + 4 p T^{5} + 108 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 161 T^{2} - 20 T^{3} + 11657 T^{4} - 20 p T^{5} + 161 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 308 T^{2} + 2936 T^{3} + 29398 T^{4} + 2936 p T^{5} + 308 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 14 T + 260 T^{2} - 2306 T^{3} + 24294 T^{4} - 2306 p T^{5} + 260 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 28 T + 541 T^{2} + 6696 T^{3} + 64817 T^{4} + 6696 p T^{5} + 541 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 18 T + 327 T^{2} + 3632 T^{3} + 36173 T^{4} + 3632 p T^{5} + 327 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 284 T^{2} + 852 T^{3} + 30822 T^{4} + 852 p T^{5} + 284 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 20 T + 445 T^{2} + 5040 T^{3} + 57977 T^{4} + 5040 p T^{5} + 445 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 212 T^{2} - 982 T^{3} + 24278 T^{4} - 982 p T^{5} + 212 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 308 T^{2} + 3096 T^{3} + 38678 T^{4} + 3096 p T^{5} + 308 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 32 T + 656 T^{2} - 8944 T^{3} + 99982 T^{4} - 8944 p T^{5} + 656 p^{2} T^{6} - 32 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.46653322256030479457133131911, −6.33161877248709105066505676088, −6.11387200296197016497719996972, −6.09453132807883582523953494758, −5.91881125975697019859783514041, −5.91228383998147132365992326564, −5.40938501291534432242309723552, −5.38917909378783571563588420637, −5.26489251187485112768707344939, −4.67653646871126466316021178592, −4.65023960387787054260264352639, −4.36717671523515481306128877683, −4.26750184037031816751411582809, −3.83762288641442262965289370583, −3.83022158982695046916981440012, −3.74438464077787755734865132368, −3.69445845662748490994219508465, −3.15238908293391198908158367679, −3.01481500789667642320806806324, −2.79913724525761956607707024079, −2.23824674346904740086739194090, −1.77563523069844661152611606034, −1.76587776140478016104067758619, −1.01568836169331341273990615428, −0.991026589383017357166346995736, 0, 0, 0, 0, 0.991026589383017357166346995736, 1.01568836169331341273990615428, 1.76587776140478016104067758619, 1.77563523069844661152611606034, 2.23824674346904740086739194090, 2.79913724525761956607707024079, 3.01481500789667642320806806324, 3.15238908293391198908158367679, 3.69445845662748490994219508465, 3.74438464077787755734865132368, 3.83022158982695046916981440012, 3.83762288641442262965289370583, 4.26750184037031816751411582809, 4.36717671523515481306128877683, 4.65023960387787054260264352639, 4.67653646871126466316021178592, 5.26489251187485112768707344939, 5.38917909378783571563588420637, 5.40938501291534432242309723552, 5.91228383998147132365992326564, 5.91881125975697019859783514041, 6.09453132807883582523953494758, 6.11387200296197016497719996972, 6.33161877248709105066505676088, 6.46653322256030479457133131911

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