Properties

Label 4-8470e2-1.1-c1e2-0-4
Degree $4$
Conductor $71740900$
Sign $1$
Analytic cond. $4574.26$
Root an. cond. $8.22394$
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  + 2·2-s + 2·3-s + 3·4-s + 2·5-s + 4·6-s − 2·7-s + 4·8-s + 4·10-s + 6·12-s − 4·13-s − 4·14-s + 4·15-s + 5·16-s + 2·19-s + 6·20-s − 4·21-s + 6·23-s + 8·24-s + 3·25-s − 8·26-s − 2·27-s − 6·28-s + 6·29-s + 8·30-s + 4·31-s + 6·32-s − 4·35-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.894·5-s + 1.63·6-s − 0.755·7-s + 1.41·8-s + 1.26·10-s + 1.73·12-s − 1.10·13-s − 1.06·14-s + 1.03·15-s + 5/4·16-s + 0.458·19-s + 1.34·20-s − 0.872·21-s + 1.25·23-s + 1.63·24-s + 3/5·25-s − 1.56·26-s − 0.384·27-s − 1.13·28-s + 1.11·29-s + 1.46·30-s + 0.718·31-s + 1.06·32-s − 0.676·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71740900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71740900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(71740900\)    =    \(2^{2} \cdot 5^{2} \cdot 7^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(4574.26\)
Root analytic conductor: \(8.22394\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{8470} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 71740900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(15.92958040\)
\(L(\frac12)\) \(\approx\) \(15.92958040\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
5$C_1$ \( ( 1 - T )^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
11 \( 1 \)
good3$D_{4}$ \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 2 T + 36 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 6 T + 52 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 6 T + 64 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 10 T + 96 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 6 T + 88 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$D_{4}$ \( 1 + 6 T + 112 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 16 T + 198 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 10 T + 180 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 12 T + 94 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 24 T + 310 T^{2} + 24 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 22 T + 288 T^{2} - 22 p T^{3} + p^{2} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.78059940453279336224313384147, −7.51602062909253101135901703858, −7.15332615899911342776755347434, −7.00521354998952660543979866252, −6.36523581560699696948527877599, −6.26015855164109240790167232095, −5.85179520421783260878237406887, −5.59995393517683811258942768870, −5.13176064360016745125757565234, −4.63933161467128895793958257258, −4.46786232377923946802825920134, −4.29422184733729054405661842238, −3.33941431028547672236942299646, −3.19090855686503637223716267846, −2.89984070631018003729540512118, −2.80843255119649005781846290588, −2.11732139466118746846439438010, −1.98340969892397530451168068999, −1.14327275344601601510968457782, −0.65572315744355252589267526058, 0.65572315744355252589267526058, 1.14327275344601601510968457782, 1.98340969892397530451168068999, 2.11732139466118746846439438010, 2.80843255119649005781846290588, 2.89984070631018003729540512118, 3.19090855686503637223716267846, 3.33941431028547672236942299646, 4.29422184733729054405661842238, 4.46786232377923946802825920134, 4.63933161467128895793958257258, 5.13176064360016745125757565234, 5.59995393517683811258942768870, 5.85179520421783260878237406887, 6.26015855164109240790167232095, 6.36523581560699696948527877599, 7.00521354998952660543979866252, 7.15332615899911342776755347434, 7.51602062909253101135901703858, 7.78059940453279336224313384147

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