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Properties

Label 4-2592e2-1.1-c1e2-0-6
Degree $4$
Conductor $6718464$
Sign $1$
Analytic cond. $428.375$
Root an. cond. $4.54942$
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  − 5-s − 2·7-s − 2·11-s − 13-s + 6·17-s + 4·19-s − 6·23-s + 5·25-s − 29-s + 8·31-s + 2·35-s + 2·37-s + 2·41-s − 10·43-s − 4·47-s + 7·49-s − 20·53-s + 2·55-s − 4·59-s − 9·61-s + 65-s + 14·67-s + 20·71-s − 18·73-s + 4·77-s − 10·79-s − 12·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.755·7-s − 0.603·11-s − 0.277·13-s + 1.45·17-s + 0.917·19-s − 1.25·23-s + 25-s − 0.185·29-s + 1.43·31-s + 0.338·35-s + 0.328·37-s + 0.312·41-s − 1.52·43-s − 0.583·47-s + 49-s − 2.74·53-s + 0.269·55-s − 0.520·59-s − 1.15·61-s + 0.124·65-s + 1.71·67-s + 2.37·71-s − 2.10·73-s + 0.455·77-s − 1.12·79-s − 1.31·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6718464\)    =    \(2^{10} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(428.375\)
Root analytic conductor: \(4.54942\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6718464,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.467439406\)
\(L(\frac12)\) \(\approx\) \(1.467439406\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
good5$C_2^2$ \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) 2.5.b_ae
7$C_2^2$ \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_ad
11$C_2^2$ \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.11.c_ah
13$C_2^2$ \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) 2.13.b_am
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.17.ag_br
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.19.ae_bq
23$C_2^2$ \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.23.g_n
29$C_2^2$ \( 1 + T - 28 T^{2} + p T^{3} + p^{2} T^{4} \) 2.29.b_abc
31$C_2^2$ \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.31.ai_bh
37$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.37.ac_cx
41$C_2^2$ \( 1 - 2 T - 37 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.41.ac_abl
43$C_2^2$ \( 1 + 10 T + 57 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.43.k_cf
47$C_2^2$ \( 1 + 4 T - 31 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.47.e_abf
53$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.53.u_hy
59$C_2^2$ \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.59.e_abr
61$C_2^2$ \( 1 + 9 T + 20 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.61.j_u
67$C_2^2$ \( 1 - 14 T + 129 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.67.ao_ez
71$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.71.au_ji
73$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.73.s_it
79$C_2^2$ \( 1 + 10 T + 21 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.79.k_v
83$C_2^2$ \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.83.m_cj
89$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.89.aw_ln
97$C_2^2$ \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.97.ac_adp
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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

−8.976776575481687436900405625741, −8.681333380154408739963357672181, −8.283555785130425759448266252831, −7.71331123089121015150988444426, −7.68492128818670605776182437801, −7.38716521883779776036574013574, −6.67172287067060160950412727455, −6.40995592567872566407449666261, −6.00108954010217990155158718778, −5.67965519100702446426703470606, −5.04100725605481669747529847434, −4.76265325387472263215621750138, −4.46490131946809158251334732399, −3.63062165590330567266322232811, −3.22388730394654678098887261393, −3.22050821594961464345686638790, −2.49779972698251651212333544661, −1.85607641557611863751098808501, −1.12868867827466942171464657597, −0.44604299955623282924723433910, 0.44604299955623282924723433910, 1.12868867827466942171464657597, 1.85607641557611863751098808501, 2.49779972698251651212333544661, 3.22050821594961464345686638790, 3.22388730394654678098887261393, 3.63062165590330567266322232811, 4.46490131946809158251334732399, 4.76265325387472263215621750138, 5.04100725605481669747529847434, 5.67965519100702446426703470606, 6.00108954010217990155158718778, 6.40995592567872566407449666261, 6.67172287067060160950412727455, 7.38716521883779776036574013574, 7.68492128818670605776182437801, 7.71331123089121015150988444426, 8.283555785130425759448266252831, 8.681333380154408739963357672181, 8.976776575481687436900405625741

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