Properties

Label 4-770e2-1.1-c1e2-0-25
Degree $4$
Conductor $592900$
Sign $1$
Analytic cond. $37.8038$
Root an. cond. $2.47961$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s − 5-s − 6-s − 5·7-s − 8-s + 3·9-s − 10-s − 11-s − 6·13-s − 5·14-s + 15-s − 16-s + 3·18-s − 6·19-s + 5·21-s − 22-s − 2·23-s + 24-s − 6·26-s − 8·27-s − 14·29-s + 30-s + 8·31-s + 33-s + 5·35-s − 6·38-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 0.447·5-s − 0.408·6-s − 1.88·7-s − 0.353·8-s + 9-s − 0.316·10-s − 0.301·11-s − 1.66·13-s − 1.33·14-s + 0.258·15-s − 1/4·16-s + 0.707·18-s − 1.37·19-s + 1.09·21-s − 0.213·22-s − 0.417·23-s + 0.204·24-s − 1.17·26-s − 1.53·27-s − 2.59·29-s + 0.182·30-s + 1.43·31-s + 0.174·33-s + 0.845·35-s − 0.973·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(592900\)    =    \(2^{2} \cdot 5^{2} \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(37.8038\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{770} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 592900,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad2$C_2$ \( 1 - T + T^{2} \)
5$C_2$ \( 1 + T + T^{2} \)
7$C_2$ \( 1 + 5 T + p T^{2} \)
11$C_2$ \( 1 + T + T^{2} \)
good3$C_2^2$ \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 2 T - 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - T - 58 T^{2} - p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
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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

−9.929900257535314900897019043432, −9.891218580170682771630753896408, −9.350274886369915167746136657465, −9.121543599353174358130377807055, −8.190839193353205938474783725430, −7.949007943436142294232167917413, −7.32971084786220971491119547725, −6.87456812144891351011368947980, −6.58403112366119352330982948527, −6.24767890824586380584268671505, −5.45703227540194634083595561453, −5.29330809832730290691945406435, −4.63596795301876302070179066248, −3.91503869310780460933049532510, −3.90366985652408114409464692162, −3.13069842121119201032603597149, −2.48213608305711172800809196996, −1.81218163680073756840792477357, 0, 0, 1.81218163680073756840792477357, 2.48213608305711172800809196996, 3.13069842121119201032603597149, 3.90366985652408114409464692162, 3.91503869310780460933049532510, 4.63596795301876302070179066248, 5.29330809832730290691945406435, 5.45703227540194634083595561453, 6.24767890824586380584268671505, 6.58403112366119352330982948527, 6.87456812144891351011368947980, 7.32971084786220971491119547725, 7.949007943436142294232167917413, 8.190839193353205938474783725430, 9.121543599353174358130377807055, 9.350274886369915167746136657465, 9.891218580170682771630753896408, 9.929900257535314900897019043432

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