Properties

Label 4-50e2-1.1-c21e2-0-4
Degree $4$
Conductor $2500$
Sign $1$
Analytic cond. $19526.8$
Root an. cond. $11.8211$
Motivic weight $21$
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  − 1.04e6·4-s + 2.04e10·9-s + 9.99e10·11-s + 1.09e12·16-s + 6.05e13·19-s + 2.33e15·29-s − 1.48e16·31-s − 2.14e16·36-s − 4.17e16·41-s − 1.04e17·44-s + 5.94e17·49-s − 4.61e18·59-s + 1.05e19·61-s − 1.15e18·64-s − 2.67e19·71-s − 6.34e19·76-s − 2.37e20·79-s + 3.08e20·81-s − 6.94e20·89-s + 2.04e21·99-s − 1.54e21·101-s + 7.84e21·109-s − 2.44e21·116-s − 7.30e21·121-s + 1.55e22·124-s + ⋯
L(s)  = 1  − 1/2·4-s + 1.95·9-s + 1.16·11-s + 1/4·16-s + 2.26·19-s + 1.03·29-s − 3.25·31-s − 0.977·36-s − 0.486·41-s − 0.580·44-s + 1.06·49-s − 1.17·59-s + 1.89·61-s − 1/8·64-s − 0.974·71-s − 1.13·76-s − 2.82·79-s + 2.81·81-s − 2.35·89-s + 2.27·99-s − 1.39·101-s + 3.17·109-s − 0.515·116-s − 0.987·121-s + 1.62·124-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s+21/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2500\)    =    \(2^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(19526.8\)
Root analytic conductor: \(11.8211\)
Motivic weight: \(21\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2500,\ (\ :21/2, 21/2),\ 1)\)

Particular Values

\(L(11)\) \(\approx\) \(4.589037580\)
\(L(\frac12)\) \(\approx\) \(4.589037580\)
\(L(\frac{23}{2})\) 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$C_2$ \( 1 + p^{20} T^{2} \)
5 \( 1 \)
good3$C_2^2$ \( 1 - 28038470 p^{6} T^{2} + p^{42} T^{4} \)
7$C_2^2$ \( 1 - 12137132054530990 p^{2} T^{2} + p^{42} T^{4} \)
11$C_2$ \( ( 1 - 49976398572 T + p^{21} T^{2} )^{2} \)
13$C_2^2$ \( 1 - \)\(49\!\cdots\!10\)\( T^{2} + p^{42} T^{4} \)
17$C_2^2$ \( 1 - \)\(10\!\cdots\!50\)\( T^{2} + p^{42} T^{4} \)
19$C_2$ \( ( 1 - 1592117156980 p T + p^{21} T^{2} )^{2} \)
23$C_2^2$ \( 1 - \)\(57\!\cdots\!10\)\( T^{2} + p^{42} T^{4} \)
29$C_2$ \( ( 1 - 1167107530943250 T + p^{21} T^{2} )^{2} \)
31$C_2$ \( ( 1 + 7431907384909648 T + p^{21} T^{2} )^{2} \)
37$C_2^2$ \( 1 + \)\(12\!\cdots\!30\)\( T^{2} + p^{42} T^{4} \)
41$C_2$ \( ( 1 + 20889593910177078 T + p^{21} T^{2} )^{2} \)
43$C_2^2$ \( 1 - \)\(34\!\cdots\!70\)\( T^{2} + p^{42} T^{4} \)
47$C_2^2$ \( 1 - \)\(99\!\cdots\!50\)\( T^{2} + p^{42} T^{4} \)
53$C_2^2$ \( 1 - \)\(14\!\cdots\!70\)\( T^{2} + p^{42} T^{4} \)
59$C_2$ \( ( 1 + 2306403035584927500 T + p^{21} T^{2} )^{2} \)
61$C_2$ \( ( 1 - 5268341017122878702 T + p^{21} T^{2} )^{2} \)
67$C_2^2$ \( 1 - \)\(32\!\cdots\!90\)\( T^{2} + p^{42} T^{4} \)
71$C_2$ \( ( 1 + 13367810994741254088 T + p^{21} T^{2} )^{2} \)
73$C_2^2$ \( 1 - \)\(23\!\cdots\!50\)\( T^{2} + p^{42} T^{4} \)
79$C_2$ \( ( 1 + \)\(11\!\cdots\!40\)\( T + p^{21} T^{2} )^{2} \)
83$C_2^2$ \( 1 - \)\(21\!\cdots\!10\)\( T^{2} + p^{42} T^{4} \)
89$C_2$ \( ( 1 + \)\(34\!\cdots\!90\)\( T + p^{21} T^{2} )^{2} \)
97$C_2^2$ \( 1 - \)\(91\!\cdots\!90\)\( T^{2} + p^{42} T^{4} \)
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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

−11.67619330162558354084496742575, −11.19477130178134744118842036793, −10.34466416286412881529845455417, −9.953544504111557726251248356989, −9.357687428841857827189815113083, −9.142283332375504660123348635321, −8.306753121544125369515706806953, −7.48989315666565868412558606718, −7.04425102208540116385835235577, −6.89412319164144661292016334381, −5.58210506596827485326192348947, −5.53126476269469219251245594784, −4.35489133392152480753343372744, −4.33665920454942775348907868998, −3.48679391756447898342699087378, −3.12953887969552196056062404792, −1.90405425739819218909386414842, −1.52492512125284861820082996786, −1.05059574562363174625931959825, −0.48711943026516728961471297607, 0.48711943026516728961471297607, 1.05059574562363174625931959825, 1.52492512125284861820082996786, 1.90405425739819218909386414842, 3.12953887969552196056062404792, 3.48679391756447898342699087378, 4.33665920454942775348907868998, 4.35489133392152480753343372744, 5.53126476269469219251245594784, 5.58210506596827485326192348947, 6.89412319164144661292016334381, 7.04425102208540116385835235577, 7.48989315666565868412558606718, 8.306753121544125369515706806953, 9.142283332375504660123348635321, 9.357687428841857827189815113083, 9.953544504111557726251248356989, 10.34466416286412881529845455417, 11.19477130178134744118842036793, 11.67619330162558354084496742575

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