Properties

Label 4-350e2-1.1-c1e2-0-1
Degree $4$
Conductor $122500$
Sign $1$
Analytic cond. $7.81070$
Root an. cond. $1.67175$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4·7-s + 8-s + 3·9-s − 3·11-s − 10·13-s + 4·14-s − 16-s + 2·17-s − 3·18-s + 5·19-s + 3·22-s + 7·23-s + 10·26-s − 8·29-s + 2·31-s − 2·34-s − 37-s − 5·38-s + 6·41-s + 4·43-s − 7·46-s + 7·47-s + 9·49-s − 9·53-s − 4·56-s + 8·58-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.51·7-s + 0.353·8-s + 9-s − 0.904·11-s − 2.77·13-s + 1.06·14-s − 1/4·16-s + 0.485·17-s − 0.707·18-s + 1.14·19-s + 0.639·22-s + 1.45·23-s + 1.96·26-s − 1.48·29-s + 0.359·31-s − 0.342·34-s − 0.164·37-s − 0.811·38-s + 0.937·41-s + 0.609·43-s − 1.03·46-s + 1.02·47-s + 9/7·49-s − 1.23·53-s − 0.534·56-s + 1.05·58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(122500\)    =    \(2^{2} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(7.81070\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 122500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5797878004\)
\(L(\frac12)\) \(\approx\) \(0.5797878004\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
good3$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
11$C_2^2$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 7 T + 26 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 7 T + 2 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 16 T + 183 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 2 T - 85 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
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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

−11.74212238638542392568587489661, −11.11584165163924703177341758099, −10.58729930652192783056899494311, −10.13474066569265854156429223107, −9.724475603072259579790942875550, −9.510857966204882455824488225416, −9.327750071305260772839371146644, −8.532657919845425121242453096632, −7.72716939365291428698653931590, −7.32382579857964788089426536430, −7.25978192377663568102751808188, −6.79614066558429530228927765220, −5.62233238596070663715367089269, −5.58120439354575535628816104719, −4.65124743374832625105413764831, −4.34548316261812174988840844362, −3.10073157014197502721780883375, −2.96673359541490778232284975198, −1.98642303142159471363102641943, −0.58332894184136697956191633983, 0.58332894184136697956191633983, 1.98642303142159471363102641943, 2.96673359541490778232284975198, 3.10073157014197502721780883375, 4.34548316261812174988840844362, 4.65124743374832625105413764831, 5.58120439354575535628816104719, 5.62233238596070663715367089269, 6.79614066558429530228927765220, 7.25978192377663568102751808188, 7.32382579857964788089426536430, 7.72716939365291428698653931590, 8.532657919845425121242453096632, 9.327750071305260772839371146644, 9.510857966204882455824488225416, 9.724475603072259579790942875550, 10.13474066569265854156429223107, 10.58729930652192783056899494311, 11.11584165163924703177341758099, 11.74212238638542392568587489661

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