Properties

Label 8-1850e4-1.1-c1e4-0-3
Degree $8$
Conductor $1.171\times 10^{13}$
Sign $1$
Analytic cond. $47620.6$
Root an. cond. $3.84347$
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  − 2·4-s + 9-s − 2·11-s + 3·16-s − 8·19-s − 6·29-s + 6·31-s − 2·36-s + 18·41-s + 4·44-s − 28·59-s − 6·61-s − 4·64-s + 24·71-s + 16·76-s + 14·79-s + 12·81-s + 8·89-s − 2·99-s + 8·101-s − 8·109-s + 12·116-s − 35·121-s − 12·124-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 4-s + 1/3·9-s − 0.603·11-s + 3/4·16-s − 1.83·19-s − 1.11·29-s + 1.07·31-s − 1/3·36-s + 2.81·41-s + 0.603·44-s − 3.64·59-s − 0.768·61-s − 1/2·64-s + 2.84·71-s + 1.83·76-s + 1.57·79-s + 4/3·81-s + 0.847·89-s − 0.201·99-s + 0.796·101-s − 0.766·109-s + 1.11·116-s − 3.18·121-s − 1.07·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 5^{8} \cdot 37^{4}\)
Sign: $1$
Analytic conductor: \(47620.6\)
Root analytic conductor: \(3.84347\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1850} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 5^{8} \cdot 37^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.691292485\)
\(L(\frac12)\) \(\approx\) \(1.691292485\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + T^{2} )^{2} \)
5 \( 1 \)
37$C_2$ \( ( 1 + T^{2} )^{2} \)
good3$C_4\times C_2$ \( 1 - T^{2} - 11 T^{4} - p^{2} T^{6} + p^{4} T^{8} \)
7$C_2^2$$\times$$C_2^2$ \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \)
11$D_{4}$ \( ( 1 + T + 19 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 45 T^{2} + 841 T^{4} - 45 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
23$D_4\times C_2$ \( 1 - 29 T^{2} + 1005 T^{4} - 29 p^{2} T^{6} + p^{4} T^{8} \)
29$C_4$ \( ( 1 + 3 T + 31 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 - 3 T + 61 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 - 9 T + 73 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 128 T^{2} + 7326 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 160 T^{2} + 10766 T^{4} - 160 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
59$D_{4}$ \( ( 1 + 14 T + 154 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 + 3 T + 43 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 45 T^{2} - 347 T^{4} - 45 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
73$D_4\times C_2$ \( 1 - 65 T^{2} + 10281 T^{4} - 65 p^{2} T^{6} + p^{4} T^{8} \)
79$D_{4}$ \( ( 1 - 7 T + 11 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 28 T^{2} - 6826 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 - 4 T + 130 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 + 36 T^{2} + 15814 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} \)
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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

−6.37916493416222010322830762330, −6.35398749140177310297830914921, −6.30087272990105668433948560233, −6.02735618369702835618610925151, −5.71868154862360284281038166984, −5.43602758158377087054906794543, −5.19516768060739279086933940796, −5.09717352272085521908028714941, −4.93171158631216274998438129249, −4.43809123102382197185587370236, −4.40597362503277572973340579017, −4.26247640333869065064156096916, −4.12974595855885795438674133190, −3.62079778364276613116429032460, −3.50602512263894727306296580840, −3.37000955572305406815820992691, −2.95965143476785906450790399701, −2.43925672926308053463527222283, −2.39386884788498814711500129560, −2.39197781983994159688185366103, −1.74670200477888196012967904422, −1.49857958825234531364959375963, −1.14046419582047499651588051164, −0.56774665187326436386963814195, −0.35897365762615882054820879967, 0.35897365762615882054820879967, 0.56774665187326436386963814195, 1.14046419582047499651588051164, 1.49857958825234531364959375963, 1.74670200477888196012967904422, 2.39197781983994159688185366103, 2.39386884788498814711500129560, 2.43925672926308053463527222283, 2.95965143476785906450790399701, 3.37000955572305406815820992691, 3.50602512263894727306296580840, 3.62079778364276613116429032460, 4.12974595855885795438674133190, 4.26247640333869065064156096916, 4.40597362503277572973340579017, 4.43809123102382197185587370236, 4.93171158631216274998438129249, 5.09717352272085521908028714941, 5.19516768060739279086933940796, 5.43602758158377087054906794543, 5.71868154862360284281038166984, 6.02735618369702835618610925151, 6.30087272990105668433948560233, 6.35398749140177310297830914921, 6.37916493416222010322830762330

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