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Properties

Label 4-425e2-1.1-c1e2-0-8
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 4·3-s + 4-s + 8·6-s + 4·7-s + 8·9-s − 8·11-s + 4·12-s + 8·14-s + 16-s + 2·17-s + 16·18-s + 16·21-s − 16·22-s + 4·23-s + 12·27-s + 4·28-s − 4·29-s − 2·32-s − 32·33-s + 4·34-s + 8·36-s + 4·37-s + 4·41-s + 32·42-s − 4·43-s − 8·44-s + ⋯
L(s)  = 1  + 1.41·2-s + 2.30·3-s + 1/2·4-s + 3.26·6-s + 1.51·7-s + 8/3·9-s − 2.41·11-s + 1.15·12-s + 2.13·14-s + 1/4·16-s + 0.485·17-s + 3.77·18-s + 3.49·21-s − 3.41·22-s + 0.834·23-s + 2.30·27-s + 0.755·28-s − 0.742·29-s − 0.353·32-s − 5.57·33-s + 0.685·34-s + 4/3·36-s + 0.657·37-s + 0.624·41-s + 4.93·42-s − 0.609·43-s − 1.20·44-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

\(L(1)\) \(\approx\) \(7.338240140\)
\(L(\frac12)\) \(\approx\) \(7.338240140\)
\(L(\frac{3}{2})\) 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$D_{4}$ \( 1 - p T + 3 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) 2.2.ac_d
3$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.3.ae_i
7$D_{4}$ \( 1 - 4 T + 16 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.7.ae_q
11$D_{4}$ \( 1 + 8 T + 36 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.11.i_bk
13$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.13.a_s
19$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.19.a_be
23$D_{4}$ \( 1 - 4 T + 48 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.23.ae_bw
29$D_{4}$ \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.29.e_cc
31$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \) 2.31.a_bs
37$D_{4}$ \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.37.ae_g
41$D_{4}$ \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.41.ae_o
43$D_{4}$ \( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.43.e_cg
47$D_{4}$ \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.47.ae_dm
53$D_{4}$ \( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.53.m_eg
59$D_{4}$ \( 1 + 24 T + 254 T^{2} + 24 p T^{3} + p^{2} T^{4} \) 2.59.y_ju
61$D_{4}$ \( 1 - 4 T + 94 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.61.ae_dq
67$D_{4}$ \( 1 - 12 T + 162 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.67.am_gg
71$C_2^2$ \( 1 + 124 T^{2} + p^{2} T^{4} \) 2.71.a_eu
73$D_{4}$ \( 1 - 4 T + 142 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.73.ae_fm
79$D_{4}$ \( 1 - 8 T + 172 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.79.ai_gq
83$D_{4}$ \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.83.ae_bq
89$D_{4}$ \( 1 + 16 T + 210 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.89.q_ic
97$D_{4}$ \( 1 - 4 T + 166 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.97.ae_gk
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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

−11.23664520772967716513190019753, −10.90577701985937228495745063114, −10.73656684521069951269535552949, −9.975485357724003750079855108179, −9.411322287712602102775235249481, −9.169353683811902248069805882531, −8.397931922459409570793110253071, −8.025131764241853419421054187004, −7.931599288782309644467158872189, −7.58492659120522843879784448101, −6.92871646654794529039133105580, −5.95927652893199961786478029183, −5.21187388188051352096779670999, −5.07832436227530072127581073966, −4.55161728048305662732947370380, −4.01125882588688955564358311101, −3.09562835530497243281441072024, −3.06745154060573483765755880299, −2.28272312873032065049441638696, −1.63782379683780994315413062874, 1.63782379683780994315413062874, 2.28272312873032065049441638696, 3.06745154060573483765755880299, 3.09562835530497243281441072024, 4.01125882588688955564358311101, 4.55161728048305662732947370380, 5.07832436227530072127581073966, 5.21187388188051352096779670999, 5.95927652893199961786478029183, 6.92871646654794529039133105580, 7.58492659120522843879784448101, 7.931599288782309644467158872189, 8.025131764241853419421054187004, 8.397931922459409570793110253071, 9.169353683811902248069805882531, 9.411322287712602102775235249481, 9.975485357724003750079855108179, 10.73656684521069951269535552949, 10.90577701985937228495745063114, 11.23664520772967716513190019753

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