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Properties

Label 4-425e2-1.1-c1e2-0-2
Degree $4$
Conductor $180625$
Sign $1$
Analytic cond. $11.5168$
Root an. cond. $1.84218$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 4-s + 2·7-s + 6·11-s + 2·12-s + 8·13-s − 3·16-s + 2·17-s + 4·19-s − 4·21-s + 6·23-s + 2·27-s − 2·28-s + 10·31-s − 12·33-s + 8·37-s − 16·39-s + 8·43-s − 6·44-s − 12·47-s + 6·48-s − 8·49-s − 4·51-s − 8·52-s − 12·53-s − 8·57-s + 12·59-s + ⋯
L(s)  = 1  − 1.15·3-s − 1/2·4-s + 0.755·7-s + 1.80·11-s + 0.577·12-s + 2.21·13-s − 3/4·16-s + 0.485·17-s + 0.917·19-s − 0.872·21-s + 1.25·23-s + 0.384·27-s − 0.377·28-s + 1.79·31-s − 2.08·33-s + 1.31·37-s − 2.56·39-s + 1.21·43-s − 0.904·44-s − 1.75·47-s + 0.866·48-s − 8/7·49-s − 0.560·51-s − 1.10·52-s − 1.64·53-s − 1.05·57-s + 1.56·59-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: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 180625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.409277323\)
\(L(\frac12)\) \(\approx\) \(1.409277323\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.2.a_b
3$D_{4}$ \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.3.c_e
7$D_{4}$ \( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.7.ac_m
11$D_{4}$ \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.11.ag_bc
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.13.ai_bq
19$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.19.ae_be
23$D_{4}$ \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_bc
29$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.29.a_bu
31$D_{4}$ \( 1 - 10 T + 84 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.31.ak_dg
37$D_{4}$ \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.37.ai_da
41$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.41.a_cs
43$D_{4}$ \( 1 - 8 T + 90 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.43.ai_dm
47$D_{4}$ \( 1 + 12 T + 82 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.47.m_de
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.53.m_fm
59$D_{4}$ \( 1 - 12 T + 142 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.59.am_fm
61$D_{4}$ \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.61.ae_da
67$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.67.au_ja
71$D_{4}$ \( 1 - 6 T + 76 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.71.ag_cy
73$D_{4}$ \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.73.ai_cc
79$D_{4}$ \( 1 + 2 T - 84 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.79.c_adg
83$D_{4}$ \( 1 + 24 T + 298 T^{2} + 24 p T^{3} + p^{2} T^{4} \) 2.83.y_lm
89$D_{4}$ \( 1 + 12 T + 106 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.89.m_ec
97$D_{4}$ \( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.97.e_fu
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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

−11.24678025274335182703938391968, −11.08053616229160852210872387872, −11.04296065343636595689534756771, −9.915611041572223019167629459422, −9.487877910666509588169201969638, −9.382188473077431894215583537339, −8.421416044887131589811179925581, −8.414091983373977338876427447773, −7.961813511297414577839190731928, −6.78996604295922861728310179721, −6.74253352063884930516399980113, −6.25265760195873851031128989022, −5.65148866906243903393819041999, −5.25317764825741969105188167366, −4.62364220456563052141525776813, −4.09315559378755493942349670022, −3.59474517355167782177825160326, −2.77588952502158478813771462910, −1.30545229801566080766941847383, −1.06565410959825641294290623053, 1.06565410959825641294290623053, 1.30545229801566080766941847383, 2.77588952502158478813771462910, 3.59474517355167782177825160326, 4.09315559378755493942349670022, 4.62364220456563052141525776813, 5.25317764825741969105188167366, 5.65148866906243903393819041999, 6.25265760195873851031128989022, 6.74253352063884930516399980113, 6.78996604295922861728310179721, 7.961813511297414577839190731928, 8.414091983373977338876427447773, 8.421416044887131589811179925581, 9.382188473077431894215583537339, 9.487877910666509588169201969638, 9.915611041572223019167629459422, 11.04296065343636595689534756771, 11.08053616229160852210872387872, 11.24678025274335182703938391968

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