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Properties

Label 4-75e2-1.1-c21e2-0-2
Degree $4$
Conductor $5625$
Sign $1$
Analytic cond. $43935.5$
Root an. cond. $14.4778$
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  + 3.89e6·4-s − 3.48e9·9-s − 1.55e11·11-s + 1.07e13·16-s + 5.72e13·19-s + 1.03e14·29-s + 1.78e16·31-s − 1.35e16·36-s + 1.16e17·41-s − 6.04e17·44-s − 5.15e17·49-s − 1.03e19·59-s + 2.50e18·61-s + 2.49e19·64-s − 2.20e19·71-s + 2.23e20·76-s − 1.26e20·79-s + 1.21e19·81-s − 2.74e20·89-s + 5.41e20·99-s + 2.29e21·101-s + 3.33e21·109-s + 4.02e20·116-s + 3.25e21·121-s + 6.95e22·124-s + 127-s + 131-s + ⋯
L(s)  = 1  + 1.85·4-s − 1/3·9-s − 1.80·11-s + 2.45·16-s + 2.14·19-s + 0.0456·29-s + 3.90·31-s − 0.619·36-s + 1.35·41-s − 3.35·44-s − 0.922·49-s − 2.62·59-s + 0.449·61-s + 2.70·64-s − 0.805·71-s + 3.97·76-s − 1.50·79-s + 1/9·81-s − 0.933·89-s + 0.601·99-s + 2.06·101-s + 1.34·109-s + 0.0847·116-s + 0.440·121-s + 7.26·124-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+21/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5625\)    =    \(3^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(43935.5\)
Root analytic conductor: \(14.4778\)
Motivic weight: \(21\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5625,\ (\ :21/2, 21/2),\ 1)\)

Particular Values

\(L(11)\) \(\approx\) \(5.856231766\)
\(L(\frac12)\) \(\approx\) \(5.856231766\)
\(L(\frac{23}{2})\) 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)$
bad3$C_2$ \( 1 + p^{20} T^{2} \)
5 \( 1 \)
good2$C_2^2$ \( 1 - 3807 p^{10} T^{2} + p^{42} T^{4} \)
7$C_2^2$ \( 1 + 10518804532409314 p^{2} T^{2} + p^{42} T^{4} \)
11$C_2$ \( ( 1 + 77585921744 T + p^{21} T^{2} )^{2} \)
13$C_2^2$ \( 1 - \)\(18\!\cdots\!38\)\( p^{2} T^{2} + p^{42} T^{4} \)
17$C_2^2$ \( 1 + \)\(26\!\cdots\!90\)\( T^{2} + p^{42} T^{4} \)
19$C_2$ \( ( 1 - 1505539797884 p T + p^{21} T^{2} )^{2} \)
23$C_2^2$ \( 1 - \)\(28\!\cdots\!46\)\( T^{2} + p^{42} T^{4} \)
29$C_2$ \( ( 1 - 51676030833142 T + p^{21} T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8921108838285000 T + p^{21} T^{2} )^{2} \)
37$C_2^2$ \( 1 + \)\(22\!\cdots\!26\)\( T^{2} + p^{42} T^{4} \)
41$C_2$ \( ( 1 - 58168090830044570 T + p^{21} T^{2} )^{2} \)
43$C_2^2$ \( 1 - \)\(14\!\cdots\!10\)\( T^{2} + p^{42} T^{4} \)
47$C_2^2$ \( 1 - \)\(23\!\cdots\!30\)\( T^{2} + p^{42} T^{4} \)
53$C_2^2$ \( 1 + \)\(20\!\cdots\!94\)\( T^{2} + p^{42} T^{4} \)
59$C_2$ \( ( 1 + 5154256088898000016 T + p^{21} T^{2} )^{2} \)
61$C_2$ \( ( 1 - 1251686105775241798 T + p^{21} T^{2} )^{2} \)
67$C_2^2$ \( 1 - \)\(41\!\cdots\!90\)\( T^{2} + p^{42} T^{4} \)
71$C_2$ \( ( 1 + 11043230850518282368 T + p^{21} T^{2} )^{2} \)
73$C_2^2$ \( 1 - \)\(12\!\cdots\!46\)\( T^{2} + p^{42} T^{4} \)
79$C_2$ \( ( 1 + 63155369968366862760 T + p^{21} T^{2} )^{2} \)
83$C_2^2$ \( 1 - \)\(18\!\cdots\!82\)\( T^{2} + p^{42} T^{4} \)
89$C_2$ \( ( 1 + \)\(13\!\cdots\!14\)\( T + p^{21} T^{2} )^{2} \)
97$C_2^2$ \( 1 - \)\(94\!\cdots\!70\)\( T^{2} + p^{42} T^{4} \)
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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

−11.38234636389406829678087864220, −10.36042220760514300015902062344, −10.03852225109877040080750869067, −9.582957060177314940491300022074, −8.541235757411729844553343037071, −7.998880043031489796676519571838, −7.63616960798431706065656080745, −7.31842652121896445789252854343, −6.56264906461255927839662847157, −6.01059813443408068690578454424, −5.75224585574992692262112503065, −4.93842359767822846108341535453, −4.59071811986835303198888610619, −3.38333037202406607495552751445, −2.95103798589061202117787510295, −2.71105110265892208653284275635, −2.30717519189310408124077141484, −1.36865520901347859720162580677, −1.10151404710942036482373667321, −0.42319798172835665039815170535, 0.42319798172835665039815170535, 1.10151404710942036482373667321, 1.36865520901347859720162580677, 2.30717519189310408124077141484, 2.71105110265892208653284275635, 2.95103798589061202117787510295, 3.38333037202406607495552751445, 4.59071811986835303198888610619, 4.93842359767822846108341535453, 5.75224585574992692262112503065, 6.01059813443408068690578454424, 6.56264906461255927839662847157, 7.31842652121896445789252854343, 7.63616960798431706065656080745, 7.998880043031489796676519571838, 8.541235757411729844553343037071, 9.582957060177314940491300022074, 10.03852225109877040080750869067, 10.36042220760514300015902062344, 11.38234636389406829678087864220

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