Properties

Label 4-8470e2-1.1-c1e2-0-0
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 − 10·19-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 − 2.29·19-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

\(L(1)\) \(\approx\) \(0.1639929298\)
\(L(\frac12)\) \(\approx\) \(0.1639929298\)
\(L(\frac{3}{2})\) 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$D_{4}$ \( 1 + 10 T + 60 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 6 T + 64 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$D_{4}$ \( 1 + 2 T + 72 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 6 T + 64 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 18 T + 184 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 14 T + 180 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 12 T + 94 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 166 T^{2} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 2 T - 48 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
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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

−7.988456251114573745021141892444, −7.962022869841743540773668008923, −7.49043061518176553786875018398, −6.91293904617056177845845610519, −6.71941525247226196483818943579, −6.52981109420041141903587523355, −6.08582358157489512443082770232, −5.84157406733271623585095954346, −4.97904534958620749140701463199, −4.78952223429476774272009523251, −4.43723244118109288391690677305, −3.81391369471345841171616337555, −3.52369994597018833594624652038, −3.19437633866738182789122358667, −2.67997533242755250837786509196, −2.46895795283752236077008714574, −2.00058586918462789122145243774, −1.65609134455063738807448029125, −0.73709488118990514894551911988, −0.14404834769734862854217692458, 0.14404834769734862854217692458, 0.73709488118990514894551911988, 1.65609134455063738807448029125, 2.00058586918462789122145243774, 2.46895795283752236077008714574, 2.67997533242755250837786509196, 3.19437633866738182789122358667, 3.52369994597018833594624652038, 3.81391369471345841171616337555, 4.43723244118109288391690677305, 4.78952223429476774272009523251, 4.97904534958620749140701463199, 5.84157406733271623585095954346, 6.08582358157489512443082770232, 6.52981109420041141903587523355, 6.71941525247226196483818943579, 6.91293904617056177845845610519, 7.49043061518176553786875018398, 7.962022869841743540773668008923, 7.988456251114573745021141892444

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