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Properties

Label 4-95e2-1.1-c1e2-0-0
Degree $4$
Conductor $9025$
Sign $1$
Analytic cond. $0.575441$
Root an. cond. $0.870964$
Motivic weight $1$
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  − 4·4-s + 3·5-s − 2·9-s + 6·11-s + 12·16-s + 2·19-s − 12·20-s + 4·25-s + 12·29-s − 8·31-s + 8·36-s − 12·41-s − 24·44-s − 6·45-s − 13·49-s + 18·55-s − 12·59-s − 2·61-s − 32·64-s + 12·71-s − 8·76-s + 16·79-s + 36·80-s − 5·81-s + 24·89-s + 6·95-s − 12·99-s + ⋯
L(s)  = 1  − 2·4-s + 1.34·5-s − 2/3·9-s + 1.80·11-s + 3·16-s + 0.458·19-s − 2.68·20-s + 4/5·25-s + 2.22·29-s − 1.43·31-s + 4/3·36-s − 1.87·41-s − 3.61·44-s − 0.894·45-s − 1.85·49-s + 2.42·55-s − 1.56·59-s − 0.256·61-s − 4·64-s + 1.42·71-s − 0.917·76-s + 1.80·79-s + 4.02·80-s − 5/9·81-s + 2.54·89-s + 0.615·95-s − 1.20·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9025\)    =    \(5^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(0.575441\)
Root analytic conductor: \(0.870964\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9025,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8268740268\)
\(L(\frac12)\) \(\approx\) \(0.8268740268\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 3 T + p T^{2} \)
19$C_1$ \( ( 1 - T )^{2} \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \)
3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
47$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
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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

−11.99055989369382628931438537309, −10.91480268940863095584089699098, −10.24320445923116880350231460909, −9.774569434898319390985970077664, −9.180168549070062598241568188033, −9.093042453849772655367942297608, −8.415277882855381679327851057927, −7.79259928613141419667093336124, −6.39084306102520193435942181609, −6.36395749589543487340848898303, −5.15254746234400988386261721371, −5.03912355415234199771433602492, −3.94480584782122104516168200313, −3.23182893022898332662250292694, −1.43009506463758030263279983978, 1.43009506463758030263279983978, 3.23182893022898332662250292694, 3.94480584782122104516168200313, 5.03912355415234199771433602492, 5.15254746234400988386261721371, 6.36395749589543487340848898303, 6.39084306102520193435942181609, 7.79259928613141419667093336124, 8.415277882855381679327851057927, 9.093042453849772655367942297608, 9.180168549070062598241568188033, 9.774569434898319390985970077664, 10.24320445923116880350231460909, 10.91480268940863095584089699098, 11.99055989369382628931438537309

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