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Properties

Label 4-425e2-1.1-c1e2-0-1
Degree $4$
Conductor $180625$
Sign $1$
Analytic cond. $11.5168$
Root an. cond. $1.84218$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 2·9-s − 3·16-s + 8·19-s + 2·36-s − 10·49-s + 24·59-s + 7·64-s − 8·76-s − 5·81-s + 12·89-s + 12·101-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 6·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 26·169-s − 16·171-s + 173-s + 179-s + ⋯
L(s)  = 1  − 1/2·4-s − 2/3·9-s − 3/4·16-s + 1.83·19-s + 1/3·36-s − 1.42·49-s + 3.12·59-s + 7/8·64-s − 0.917·76-s − 5/9·81-s + 1.27·89-s + 1.19·101-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/2·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2·169-s − 1.22·171-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 180625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(180625\)    =    \(5^{4} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(11.5168\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 180625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.237362722\)
\(L(\frac12)\) \(\approx\) \(1.237362722\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad5 \( 1 \)
17$C_2$ \( 1 + p T^{2} \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.2.a_b
3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.3.a_c
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.11.a_ak
13$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.13.a_ba
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.19.ai_cc
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.29.a_ak
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.37.a_cg
41$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.41.a_abi
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.43.a_aw
47$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \) 2.47.a_de
53$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \) 2.53.a_cg
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.59.ay_kc
61$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.61.a_aes
67$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.67.a_ba
71$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.71.a_abi
73$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.73.a_fa
79$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.79.a_aby
83$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \) 2.83.a_cg
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.89.am_ig
97$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.97.a_fa
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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

−9.074541137019956347885241111637, −8.791950739576814931072123892726, −8.222147702646035633564079509111, −7.81950112317260494645801119062, −7.20201007608899037810626173729, −6.81398874796311922894451571339, −6.19359996905995423396835479517, −5.55173467474692713066630399667, −5.19671510295305426214108380883, −4.68783216819253852818427846160, −3.96402871953427029927071990295, −3.36584666064233594496411613345, −2.77589417464022821809423785733, −1.92680040076152715471852041288, −0.73300880709545938079257413225, 0.73300880709545938079257413225, 1.92680040076152715471852041288, 2.77589417464022821809423785733, 3.36584666064233594496411613345, 3.96402871953427029927071990295, 4.68783216819253852818427846160, 5.19671510295305426214108380883, 5.55173467474692713066630399667, 6.19359996905995423396835479517, 6.81398874796311922894451571339, 7.20201007608899037810626173729, 7.81950112317260494645801119062, 8.222147702646035633564079509111, 8.791950739576814931072123892726, 9.074541137019956347885241111637

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