Properties

Label 4-5e4-1.1-c25e2-0-0
Degree $4$
Conductor $625$
Sign $1$
Analytic cond. $9800.84$
Root an. cond. $9.94983$
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  + 6.71e7·4-s + 1.65e12·9-s + 1.68e13·11-s + 3.37e15·16-s + 1.21e16·19-s + 5.42e17·29-s + 8.58e18·31-s + 1.11e20·36-s − 3.67e20·41-s + 1.13e21·44-s + 1.15e21·49-s − 2.61e22·59-s + 1.80e22·61-s + 1.51e23·64-s − 3.84e23·71-s + 8.16e23·76-s + 5.43e23·79-s + 2.02e24·81-s + 3.52e24·89-s + 2.78e25·99-s + 3.72e24·101-s + 9.55e25·109-s + 3.64e25·116-s − 4.02e24·121-s + 5.75e26·124-s + 127-s + 131-s + ⋯
L(s)  = 1  + 1.99·4-s + 1.95·9-s + 1.61·11-s + 2.99·16-s + 1.26·19-s + 0.284·29-s + 1.95·31-s + 3.90·36-s − 2.54·41-s + 3.23·44-s + 0.861·49-s − 1.91·59-s + 0.869·61-s + 3.99·64-s − 2.78·71-s + 2.52·76-s + 1.03·79-s + 2.82·81-s + 1.51·89-s + 3.16·99-s + 0.329·101-s + 3.25·109-s + 0.569·116-s − 0.0371·121-s + 3.91·124-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+25/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(625\)    =    \(5^{4}\)
Sign: $1$
Analytic conductor: \(9800.84\)
Root analytic conductor: \(9.94983\)
Motivic weight: \(25\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 625,\ (\ :25/2, 25/2),\ 1)\)

Particular Values

\(L(13)\) \(\approx\) \(13.21208365\)
\(L(\frac12)\) \(\approx\) \(13.21208365\)
\(L(\frac{27}{2})\) 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)$
bad5 \( 1 \)
good2$C_2^2$ \( 1 - 262135 p^{8} T^{2} + p^{50} T^{4} \)
3$C_2^2$ \( 1 - 2271931430 p^{6} T^{2} + p^{50} T^{4} \)
7$C_2^2$ \( 1 - 480984657411316750 p^{4} T^{2} + p^{50} T^{4} \)
11$C_2$ \( ( 1 - 765410481732 p T + p^{25} T^{2} )^{2} \)
13$C_2^2$ \( 1 - \)\(44\!\cdots\!10\)\( p^{2} T^{2} + p^{50} T^{4} \)
17$C_2^2$ \( 1 - \)\(17\!\cdots\!70\)\( p^{2} T^{2} + p^{50} T^{4} \)
19$C_2$ \( ( 1 - 320108230016260 p T + p^{25} T^{2} )^{2} \)
23$C_2^2$ \( 1 - \)\(24\!\cdots\!90\)\( p^{2} T^{2} + p^{50} T^{4} \)
29$C_2$ \( ( 1 - 271246959476737410 T + p^{25} T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4291666067521509152 T + p^{25} T^{2} )^{2} \)
37$C_2^2$ \( 1 - \)\(27\!\cdots\!90\)\( T^{2} + p^{50} T^{4} \)
41$C_2$ \( ( 1 + \)\(18\!\cdots\!98\)\( T + p^{25} T^{2} )^{2} \)
43$C_2^2$ \( 1 - \)\(46\!\cdots\!50\)\( T^{2} + p^{50} T^{4} \)
47$C_2^2$ \( 1 - \)\(41\!\cdots\!70\)\( T^{2} + p^{50} T^{4} \)
53$C_2^2$ \( 1 - \)\(24\!\cdots\!70\)\( T^{2} + p^{50} T^{4} \)
59$C_2$ \( ( 1 + \)\(13\!\cdots\!80\)\( T + p^{25} T^{2} )^{2} \)
61$C_2$ \( ( 1 - \)\(90\!\cdots\!02\)\( T + p^{25} T^{2} )^{2} \)
67$C_2^2$ \( 1 - \)\(82\!\cdots\!30\)\( T^{2} + p^{50} T^{4} \)
71$C_2$ \( ( 1 + \)\(19\!\cdots\!48\)\( T + p^{25} T^{2} )^{2} \)
73$C_2^2$ \( 1 - \)\(74\!\cdots\!10\)\( T^{2} + p^{50} T^{4} \)
79$C_2$ \( ( 1 - \)\(27\!\cdots\!60\)\( T + p^{25} T^{2} )^{2} \)
83$C_2^2$ \( 1 - \)\(10\!\cdots\!30\)\( T^{2} + p^{50} T^{4} \)
89$C_2$ \( ( 1 - \)\(17\!\cdots\!30\)\( T + p^{25} T^{2} )^{2} \)
97$C_2^2$ \( 1 - \)\(85\!\cdots\!70\)\( T^{2} + p^{50} T^{4} \)
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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

−12.12008900932182728886820474660, −12.03743822382065575559198500553, −11.64666502029399476707814223085, −10.75999455121403711229415765039, −10.03104510430905611961860172574, −10.00899364632405505313022878538, −8.997168779348518884236944855081, −8.050155962195873998427181785333, −7.33676007093481220678349713540, −7.08057749658177703561591762160, −6.42067115473990850310971331411, −6.16228919840969828852366202382, −5.04247148704371041283838971388, −4.35203390620115321510580939190, −3.50987995831921873148194917578, −3.15836819733434756394781710690, −2.20871615100511393397996153632, −1.60779795351867689517147118310, −1.24401322257868636129892952772, −0.839110201010052614100169168926, 0.839110201010052614100169168926, 1.24401322257868636129892952772, 1.60779795351867689517147118310, 2.20871615100511393397996153632, 3.15836819733434756394781710690, 3.50987995831921873148194917578, 4.35203390620115321510580939190, 5.04247148704371041283838971388, 6.16228919840969828852366202382, 6.42067115473990850310971331411, 7.08057749658177703561591762160, 7.33676007093481220678349713540, 8.050155962195873998427181785333, 8.997168779348518884236944855081, 10.00899364632405505313022878538, 10.03104510430905611961860172574, 10.75999455121403711229415765039, 11.64666502029399476707814223085, 12.03743822382065575559198500553, 12.12008900932182728886820474660

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