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Properties

Label 4-50e2-1.1-c25e2-0-1
Degree $4$
Conductor $2500$
Sign $1$
Analytic cond. $39203.3$
Root an. cond. $14.0711$
Motivic weight $25$
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  − 1.67e7·4-s + 1.66e12·9-s − 2.24e13·11-s + 2.81e14·16-s − 5.97e15·19-s + 4.24e18·29-s + 8.45e18·31-s − 2.79e19·36-s − 3.23e20·41-s + 3.77e20·44-s + 2.37e21·49-s − 2.42e22·59-s − 4.06e22·61-s − 4.72e21·64-s + 2.68e23·71-s + 1.00e23·76-s − 7.64e23·79-s + 2.06e24·81-s + 1.68e24·89-s − 3.74e25·99-s + 1.32e25·101-s − 9.72e25·109-s − 7.11e25·116-s + 1.62e26·121-s − 1.41e26·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 1/2·4-s + 1.96·9-s − 2.15·11-s + 1/4·16-s − 0.619·19-s + 2.22·29-s + 1.92·31-s − 0.984·36-s − 2.23·41-s + 1.07·44-s + 1.76·49-s − 1.77·59-s − 1.96·61-s − 1/8·64-s + 1.93·71-s + 0.309·76-s − 1.45·79-s + 2.87·81-s + 0.721·89-s − 4.25·99-s + 1.17·101-s − 3.31·109-s − 1.11·116-s + 1.49·121-s − 0.963·124-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s+25/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2500\)    =    \(2^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(39203.3\)
Root analytic conductor: \(14.0711\)
Motivic weight: \(25\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2500,\ (\ :25/2, 25/2),\ 1)\)

Particular Values

\(L(13)\) \(\approx\) \(2.875008008\)
\(L(\frac12)\) \(\approx\) \(2.875008008\)
\(L(\frac{27}{2})\) 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$C_2$ \( 1 + p^{24} T^{2} \)
5 \( 1 \)
good3$C_2^2$ \( 1 - 2288137910 p^{6} T^{2} + p^{50} T^{4} \)
7$C_2^2$ \( 1 - 988065717455727550 p^{4} T^{2} + p^{50} T^{4} \)
11$C_2$ \( ( 1 + 1021852166868 p T + p^{25} T^{2} )^{2} \)
13$C_2^2$ \( 1 - \)\(14\!\cdots\!30\)\( T^{2} + p^{50} T^{4} \)
17$C_2^2$ \( 1 - \)\(10\!\cdots\!10\)\( T^{2} + p^{50} T^{4} \)
19$C_2$ \( ( 1 + 157360283511980 p T + p^{25} T^{2} )^{2} \)
23$C_2^2$ \( 1 - \)\(13\!\cdots\!30\)\( p^{2} T^{2} + p^{50} T^{4} \)
29$C_2$ \( ( 1 - 2120475579683207970 T + p^{25} T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4225863091688971352 T + p^{25} T^{2} )^{2} \)
37$C_2^2$ \( 1 - \)\(26\!\cdots\!30\)\( T^{2} + p^{50} T^{4} \)
41$C_2$ \( ( 1 + \)\(16\!\cdots\!98\)\( T + p^{25} T^{2} )^{2} \)
43$C_2^2$ \( 1 - \)\(40\!\cdots\!50\)\( T^{2} + p^{50} T^{4} \)
47$C_2^2$ \( 1 + \)\(23\!\cdots\!10\)\( T^{2} + p^{50} T^{4} \)
53$C_2^2$ \( 1 - \)\(25\!\cdots\!90\)\( T^{2} + p^{50} T^{4} \)
59$C_2$ \( ( 1 + \)\(12\!\cdots\!60\)\( T + p^{25} T^{2} )^{2} \)
61$C_2$ \( ( 1 + \)\(20\!\cdots\!98\)\( T + p^{25} T^{2} )^{2} \)
67$C_2^2$ \( 1 - \)\(51\!\cdots\!10\)\( T^{2} + p^{50} T^{4} \)
71$C_2$ \( ( 1 - \)\(13\!\cdots\!52\)\( T + p^{25} T^{2} )^{2} \)
73$C_2^2$ \( 1 - \)\(26\!\cdots\!70\)\( T^{2} + p^{50} T^{4} \)
79$C_2$ \( ( 1 + \)\(38\!\cdots\!80\)\( T + p^{25} T^{2} )^{2} \)
83$C_2^2$ \( 1 - \)\(13\!\cdots\!10\)\( T^{2} + p^{50} T^{4} \)
89$C_2$ \( ( 1 - \)\(84\!\cdots\!10\)\( T + p^{25} T^{2} )^{2} \)
97$C_2^2$ \( 1 - \)\(55\!\cdots\!90\)\( T^{2} + p^{50} T^{4} \)
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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

−10.69153528165787971105627668688, −10.50689703299630834892066667024, −10.15417686533940605923419673761, −9.726858456026688443134010276292, −8.887756914342835971782710552495, −8.276248078726072387486559006916, −7.86878182171569829002187791916, −7.42729367330955195762598628777, −6.54124816270118636361090111646, −6.43271720711638530179298423396, −5.20602737546580880107925699312, −5.06925420225393419449581586950, −4.31385626968626866628725584132, −4.20696234952478921706195446948, −2.97633327446984490235084131450, −2.89091712956766744060535347024, −1.99708276503993726520539609957, −1.49527397258355509241562272706, −0.78418962322735280677430921680, −0.40398329640337315461319682551, 0.40398329640337315461319682551, 0.78418962322735280677430921680, 1.49527397258355509241562272706, 1.99708276503993726520539609957, 2.89091712956766744060535347024, 2.97633327446984490235084131450, 4.20696234952478921706195446948, 4.31385626968626866628725584132, 5.06925420225393419449581586950, 5.20602737546580880107925699312, 6.43271720711638530179298423396, 6.54124816270118636361090111646, 7.42729367330955195762598628777, 7.86878182171569829002187791916, 8.276248078726072387486559006916, 8.887756914342835971782710552495, 9.726858456026688443134010276292, 10.15417686533940605923419673761, 10.50689703299630834892066667024, 10.69153528165787971105627668688

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