Properties

Label 4-50e2-1.1-c21e2-0-0
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 + 1.74e10·9-s − 2.13e11·11-s + 1.09e12·16-s − 2.20e13·19-s − 4.76e15·29-s − 1.75e15·31-s − 1.82e16·36-s − 4.92e16·41-s + 2.23e17·44-s − 9.20e17·49-s + 5.91e18·59-s + 1.59e19·61-s − 1.15e18·64-s + 1.76e19·71-s + 2.31e19·76-s − 6.76e19·79-s + 1.93e20·81-s + 8.20e19·89-s − 3.71e21·99-s + 1.18e21·101-s − 4.03e21·109-s + 4.99e21·116-s + 1.93e22·121-s + 1.84e21·124-s + ⋯
L(s)  = 1  − 1/2·4-s + 1.66·9-s − 2.48·11-s + 1/4·16-s − 0.825·19-s − 2.10·29-s − 0.385·31-s − 0.831·36-s − 0.572·41-s + 1.24·44-s − 1.64·49-s + 1.50·59-s + 2.86·61-s − 1/8·64-s + 0.645·71-s + 0.412·76-s − 0.804·79-s + 1.76·81-s + 0.278·89-s − 4.12·99-s + 1.07·101-s − 1.63·109-s + 1.05·116-s + 2.62·121-s + 0.192·124-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

\(L(11)\) \(\approx\) \(1.286340496\)
\(L(\frac12)\) \(\approx\) \(1.286340496\)
\(L(\frac{23}{2})\) 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$C_2^2$ \( 1 - 1933590950 p^{2} T^{2} + p^{42} T^{4} \)
7$C_2^2$ \( 1 + 18784750565190290 p^{2} T^{2} + p^{42} T^{4} \)
11$C_2$ \( ( 1 + 9706172268 p T + p^{21} T^{2} )^{2} \)
13$C_2^2$ \( 1 - \)\(27\!\cdots\!50\)\( p^{2} T^{2} + p^{42} T^{4} \)
17$C_2^2$ \( 1 - \)\(43\!\cdots\!30\)\( p^{2} T^{2} + p^{42} T^{4} \)
19$C_2$ \( ( 1 + 580213471340 p T + p^{21} T^{2} )^{2} \)
23$C_2^2$ \( 1 - \)\(62\!\cdots\!30\)\( T^{2} + p^{42} T^{4} \)
29$C_2$ \( ( 1 + 2382370826608110 T + p^{21} T^{2} )^{2} \)
31$C_2$ \( ( 1 + 878552957377888 T + p^{21} T^{2} )^{2} \)
37$C_2^2$ \( 1 - \)\(74\!\cdots\!90\)\( T^{2} + p^{42} T^{4} \)
41$C_2$ \( ( 1 + 24612925945718838 T + p^{21} T^{2} )^{2} \)
43$C_2^2$ \( 1 - \)\(22\!\cdots\!30\)\( T^{2} + p^{42} T^{4} \)
47$C_2^2$ \( 1 - \)\(22\!\cdots\!30\)\( T^{2} + p^{42} T^{4} \)
53$C_2^2$ \( 1 - \)\(28\!\cdots\!10\)\( T^{2} + p^{42} T^{4} \)
59$C_2$ \( ( 1 - 2955954134483673780 T + p^{21} T^{2} )^{2} \)
61$C_2$ \( ( 1 - 7984150090052846222 T + p^{21} T^{2} )^{2} \)
67$C_2^2$ \( 1 - \)\(42\!\cdots\!30\)\( T^{2} + p^{42} T^{4} \)
71$C_2$ \( ( 1 - 8849017338933008232 T + p^{21} T^{2} )^{2} \)
73$C_2^2$ \( 1 - \)\(13\!\cdots\!90\)\( T^{2} + p^{42} T^{4} \)
79$C_2$ \( ( 1 + 33840609578636773520 T + p^{21} T^{2} )^{2} \)
83$C_2^2$ \( 1 + \)\(17\!\cdots\!90\)\( T^{2} + p^{42} T^{4} \)
89$C_2$ \( ( 1 - 41024056743692272710 T + p^{21} T^{2} )^{2} \)
97$C_2^2$ \( 1 - \)\(52\!\cdots\!90\)\( T^{2} + p^{42} T^{4} \)
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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

−11.58853654263874544021090680266, −10.93630812341661079768839303231, −10.51061021540967764248500975055, −9.886596085472561051499759243879, −9.728506530426118375115913930014, −8.823304205727543437255640842190, −8.084256301051287213614922015362, −7.84732627649111512030196500606, −7.12176441442602752759265731693, −6.73040802983817415100185122221, −5.62512160876956651533427933687, −5.32638353637116988902662087435, −4.75681416305888113936882787643, −4.09982424601330285745144594225, −3.60778714657131260154139789276, −2.82967795509440754675334941170, −1.92488484103355216257403409450, −1.92073137547734546886911275297, −0.78606570688066628316651994737, −0.28836754703506410747344116589, 0.28836754703506410747344116589, 0.78606570688066628316651994737, 1.92073137547734546886911275297, 1.92488484103355216257403409450, 2.82967795509440754675334941170, 3.60778714657131260154139789276, 4.09982424601330285745144594225, 4.75681416305888113936882787643, 5.32638353637116988902662087435, 5.62512160876956651533427933687, 6.73040802983817415100185122221, 7.12176441442602752759265731693, 7.84732627649111512030196500606, 8.084256301051287213614922015362, 8.823304205727543437255640842190, 9.728506530426118375115913930014, 9.886596085472561051499759243879, 10.51061021540967764248500975055, 10.93630812341661079768839303231, 11.58853654263874544021090680266

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