Properties

Label 8-1815e4-1.1-c1e4-0-9
Degree $8$
Conductor $1.085\times 10^{13}$
Sign $1$
Analytic cond. $44117.9$
Root an. cond. $3.80694$
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  − 5·2-s + 4·3-s + 13·4-s − 4·5-s − 20·6-s + 2·7-s − 25·8-s + 10·9-s + 20·10-s + 52·12-s − 3·13-s − 10·14-s − 16·15-s + 40·16-s − 20·17-s − 50·18-s − 3·19-s − 52·20-s + 8·21-s − 5·23-s − 100·24-s + 10·25-s + 15·26-s + 20·27-s + 26·28-s − 5·29-s + 80·30-s + ⋯
L(s)  = 1  − 3.53·2-s + 2.30·3-s + 13/2·4-s − 1.78·5-s − 8.16·6-s + 0.755·7-s − 8.83·8-s + 10/3·9-s + 6.32·10-s + 15.0·12-s − 0.832·13-s − 2.67·14-s − 4.13·15-s + 10·16-s − 4.85·17-s − 11.7·18-s − 0.688·19-s − 11.6·20-s + 1.74·21-s − 1.04·23-s − 20.4·24-s + 2·25-s + 2.94·26-s + 3.84·27-s + 4.91·28-s − 0.928·29-s + 14.6·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 5^{4} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(44117.9\)
Root analytic conductor: \(3.80694\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1815} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 3^{4} \cdot 5^{4} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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$C_1$ \( ( 1 - T )^{4} \)
5$C_1$ \( ( 1 + T )^{4} \)
11 \( 1 \)
good2$C_2^2:C_4$ \( 1 + 5 T + 3 p^{2} T^{2} + 5 p^{2} T^{3} + 29 T^{4} + 5 p^{3} T^{5} + 3 p^{4} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 12 T^{2} - 25 T^{3} + 111 T^{4} - 25 p T^{5} + 12 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 3 T + 42 T^{2} + 123 T^{3} + 753 T^{4} + 123 p T^{5} + 42 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2$ \( ( 1 + 5 T + p T^{2} )^{4} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 3 T + 26 T^{2} - 33 T^{3} + 235 T^{4} - 33 p T^{5} + 26 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 + 5 T + 91 T^{2} + 340 T^{3} + 3127 T^{4} + 340 p T^{5} + 91 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 5 T + 72 T^{2} + 295 T^{3} + 3033 T^{4} + 295 p T^{5} + 72 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + T + 99 T^{2} + 86 T^{3} + 4355 T^{4} + 86 p T^{5} + 99 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 7 T + 3 p T^{2} + 644 T^{3} + 5907 T^{4} + 644 p T^{5} + 3 p^{3} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 20 T + 250 T^{2} + 2125 T^{3} + 15067 T^{4} + 2125 p T^{5} + 250 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 80 T^{2} - 195 T^{3} + 5043 T^{4} - 195 p T^{5} + 80 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 20 T + 6 p T^{2} + 2795 T^{3} + 22079 T^{4} + 2795 p T^{5} + 6 p^{3} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 112 T^{2} - 177 T^{3} + 4983 T^{4} - 177 p T^{5} + 112 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 5 T + 135 T^{2} + 560 T^{3} + 11267 T^{4} + 560 p T^{5} + 135 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 7 T + 108 T^{2} - 959 T^{3} + 125 p T^{4} - 959 p T^{5} + 108 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 13 T + 132 T^{2} + 845 T^{3} + 5331 T^{4} + 845 p T^{5} + 132 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 25 T + 455 T^{2} + 5300 T^{3} + 52177 T^{4} + 5300 p T^{5} + 455 p^{2} T^{6} + 25 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 23 T + 340 T^{2} + 3315 T^{3} + 29783 T^{4} + 3315 p T^{5} + 340 p^{2} T^{6} + 23 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 207 T^{2} - 210 T^{3} + 21423 T^{4} - 210 p T^{5} + 207 p^{2} T^{6} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 33 T + 597 T^{2} + 7348 T^{3} + 73103 T^{4} + 7348 p T^{5} + 597 p^{2} T^{6} + 33 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 401 T^{2} - 4140 T^{3} + 55265 T^{4} - 4140 p T^{5} + 401 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 328 T^{2} + 125 T^{3} + 44789 T^{4} + 125 p T^{5} + 328 p^{2} T^{6} + p^{4} T^{8} \)
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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

−7.08139534321304596693048304990, −7.00573758078770365782765787161, −6.91887833325000422313581293837, −6.85380926165783537258127240312, −6.44098619685757026685400425280, −6.19086467283757828296315888362, −6.15285424256003819915166534781, −5.50063552856583888126251069102, −5.42137173514394006587323050218, −4.70991941834371149458067458686, −4.66484749668547973980680370742, −4.54959977968657560593957244979, −4.46610126456342554425572227736, −4.20214309889305666672614843019, −3.82381326750442774592669024989, −3.43124774553102377367010622118, −3.31114649945115415275078150721, −3.19384974760876895743772796387, −2.70371630955836292923015196050, −2.42438114608179200841232967448, −2.27148852722030168061340599190, −2.03613793703779452970580103067, −1.64996811205198278011163227357, −1.56809857844878514586786121706, −1.32401833821696226416551924500, 0, 0, 0, 0, 1.32401833821696226416551924500, 1.56809857844878514586786121706, 1.64996811205198278011163227357, 2.03613793703779452970580103067, 2.27148852722030168061340599190, 2.42438114608179200841232967448, 2.70371630955836292923015196050, 3.19384974760876895743772796387, 3.31114649945115415275078150721, 3.43124774553102377367010622118, 3.82381326750442774592669024989, 4.20214309889305666672614843019, 4.46610126456342554425572227736, 4.54959977968657560593957244979, 4.66484749668547973980680370742, 4.70991941834371149458067458686, 5.42137173514394006587323050218, 5.50063552856583888126251069102, 6.15285424256003819915166534781, 6.19086467283757828296315888362, 6.44098619685757026685400425280, 6.85380926165783537258127240312, 6.91887833325000422313581293837, 7.00573758078770365782765787161, 7.08139534321304596693048304990

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