Properties

Label 4-8470e2-1.1-c1e2-0-2
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 − 3-s + 3·4-s − 2·5-s + 2·6-s + 2·7-s − 4·8-s − 4·9-s + 4·10-s − 3·12-s − 6·13-s − 4·14-s + 2·15-s + 5·16-s − 17-s + 8·18-s − 3·19-s − 6·20-s − 2·21-s − 8·23-s + 4·24-s + 3·25-s + 12·26-s + 6·27-s + 6·28-s + 8·29-s − 4·30-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s + 3/2·4-s − 0.894·5-s + 0.816·6-s + 0.755·7-s − 1.41·8-s − 4/3·9-s + 1.26·10-s − 0.866·12-s − 1.66·13-s − 1.06·14-s + 0.516·15-s + 5/4·16-s − 0.242·17-s + 1.88·18-s − 0.688·19-s − 1.34·20-s − 0.436·21-s − 1.66·23-s + 0.816·24-s + 3/5·25-s + 2.35·26-s + 1.15·27-s + 1.13·28-s + 1.48·29-s − 0.730·30-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

\(L(1)\) \(\approx\) \(0.4323845716\)
\(L(\frac12)\) \(\approx\) \(0.4323845716\)
\(L(\frac{3}{2})\) 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$D_{4}$ \( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + T + 33 T^{2} + p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 3 T + 39 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 42 T^{2} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 15 T + 137 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 7 T + 97 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 18 T + 170 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 5 T + 123 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 + T + 103 T^{2} + p T^{3} + p^{2} T^{4} \)
71$C_4$ \( 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 7 T + 97 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 10 T + 138 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 11 T + 165 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 13 T + 209 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 3 T + 45 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
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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

−7.86313493333606840239431088033, −7.78433942543069273731384899382, −7.47048303237400998249831321071, −7.16085862759889654441995851646, −6.45905101675920342266162101569, −6.36565943703155767575419982657, −5.94523920042829001315141462800, −5.90185869858026475314808896363, −4.96036251354591924905864099665, −4.93017822102945869922969412424, −4.52213184021877608414672465847, −4.28481647097184296036985897275, −3.37368447540678475986188033663, −3.27500799836459067882814022239, −2.55700424230590198328496325820, −2.50446520073044682072894822147, −1.87309826651203929108742451742, −1.45370987193712808823075142134, −0.46805817181426571623948303337, −0.42389219292443958376726630680, 0.42389219292443958376726630680, 0.46805817181426571623948303337, 1.45370987193712808823075142134, 1.87309826651203929108742451742, 2.50446520073044682072894822147, 2.55700424230590198328496325820, 3.27500799836459067882814022239, 3.37368447540678475986188033663, 4.28481647097184296036985897275, 4.52213184021877608414672465847, 4.93017822102945869922969412424, 4.96036251354591924905864099665, 5.90185869858026475314808896363, 5.94523920042829001315141462800, 6.36565943703155767575419982657, 6.45905101675920342266162101569, 7.16085862759889654441995851646, 7.47048303237400998249831321071, 7.78433942543069273731384899382, 7.86313493333606840239431088033

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