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Properties

Label 4-3040e2-1.1-c1e2-0-1
Degree $4$
Conductor $9241600$
Sign $1$
Analytic cond. $589.252$
Root an. cond. $4.92691$
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  + 2·5-s − 6·9-s − 8·11-s + 4·13-s + 4·17-s + 2·19-s + 3·25-s + 4·29-s + 4·37-s + 4·41-s − 12·45-s + 16·47-s − 6·49-s + 20·53-s − 16·55-s + 8·59-s + 12·61-s + 8·65-s + 20·73-s + 27·81-s + 16·83-s + 8·85-s − 12·89-s + 4·95-s + 20·97-s + 48·99-s + 28·101-s + ⋯
L(s)  = 1  + 0.894·5-s − 2·9-s − 2.41·11-s + 1.10·13-s + 0.970·17-s + 0.458·19-s + 3/5·25-s + 0.742·29-s + 0.657·37-s + 0.624·41-s − 1.78·45-s + 2.33·47-s − 6/7·49-s + 2.74·53-s − 2.15·55-s + 1.04·59-s + 1.53·61-s + 0.992·65-s + 2.34·73-s + 3·81-s + 1.75·83-s + 0.867·85-s − 1.27·89-s + 0.410·95-s + 2.03·97-s + 4.82·99-s + 2.78·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9241600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9241600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9241600\)    =    \(2^{10} \cdot 5^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(589.252\)
Root analytic conductor: \(4.92691\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9241600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.828470458\)
\(L(\frac12)\) \(\approx\) \(2.828470458\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad2 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
19$C_1$ \( ( 1 - T )^{2} \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.3.a_g
7$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.7.a_g
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.11.i_bm
13$D_{4}$ \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.13.ae_w
17$C_4$ \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.17.ae_g
23$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.23.a_bm
29$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.29.ae_be
31$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.31.a_be
37$D_{4}$ \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.37.ae_g
41$D_{4}$ \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.41.ae_cc
43$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.43.a_o
47$D_{4}$ \( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.47.aq_fu
53$D_{4}$ \( 1 - 20 T + 198 T^{2} - 20 p T^{3} + p^{2} T^{4} \) 2.53.au_hq
59$D_{4}$ \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.59.ai_dy
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.61.am_gc
67$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.67.a_g
71$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \) 2.71.a_eg
73$C_4$ \( 1 - 20 T + 214 T^{2} - 20 p T^{3} + p^{2} T^{4} \) 2.73.au_ig
79$C_2^2$ \( 1 + 126 T^{2} + p^{2} T^{4} \) 2.79.a_ew
83$D_{4}$ \( 1 - 16 T + 222 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.83.aq_io
89$D_{4}$ \( 1 + 12 T + 182 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.89.m_ha
97$D_{4}$ \( 1 - 20 T + 286 T^{2} - 20 p T^{3} + p^{2} T^{4} \) 2.97.au_la
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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

−8.813703999889958891817843327181, −8.453142250542969947469072662704, −8.269630403923437602814320268165, −7.88687597758147769144736532226, −7.46370257903597160715430393542, −7.16455071887969581342694559127, −6.31154733332721013224236341089, −6.30100053950172183370289238917, −5.63359198611362560931903279564, −5.56066276964486856954441836537, −5.22752817180897228256460995701, −4.98704739326636687552497172350, −4.16291301464604749526679732908, −3.59289119696118174974820832810, −3.24817228578289059087291272686, −2.68349065370943338172867926363, −2.35904406236073457172573558421, −2.19419972242979085508881997823, −0.831711931613628138610284796756, −0.72470752682373008436365723487, 0.72470752682373008436365723487, 0.831711931613628138610284796756, 2.19419972242979085508881997823, 2.35904406236073457172573558421, 2.68349065370943338172867926363, 3.24817228578289059087291272686, 3.59289119696118174974820832810, 4.16291301464604749526679732908, 4.98704739326636687552497172350, 5.22752817180897228256460995701, 5.56066276964486856954441836537, 5.63359198611362560931903279564, 6.30100053950172183370289238917, 6.31154733332721013224236341089, 7.16455071887969581342694559127, 7.46370257903597160715430393542, 7.88687597758147769144736532226, 8.269630403923437602814320268165, 8.453142250542969947469072662704, 8.813703999889958891817843327181

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