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Properties

Label 4-2340e2-1.1-c1e2-0-14
Degree $4$
Conductor $5475600$
Sign $1$
Analytic cond. $349.129$
Root an. cond. $4.32261$
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  + 2·5-s + 7-s + 3·11-s + 2·13-s − 3·17-s − 5·19-s + 9·23-s + 3·25-s − 9·29-s + 16·31-s + 2·35-s + 7·37-s + 3·41-s + 43-s + 7·49-s − 12·53-s + 6·55-s + 9·59-s + 61-s + 4·65-s − 5·67-s + 9·71-s + 4·73-s + 3·77-s + 16·79-s − 6·85-s + 3·89-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.377·7-s + 0.904·11-s + 0.554·13-s − 0.727·17-s − 1.14·19-s + 1.87·23-s + 3/5·25-s − 1.67·29-s + 2.87·31-s + 0.338·35-s + 1.15·37-s + 0.468·41-s + 0.152·43-s + 49-s − 1.64·53-s + 0.809·55-s + 1.17·59-s + 0.128·61-s + 0.496·65-s − 0.610·67-s + 1.06·71-s + 0.468·73-s + 0.341·77-s + 1.80·79-s − 0.650·85-s + 0.317·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5475600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5475600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5475600\)    =    \(2^{4} \cdot 3^{4} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(349.129\)
Root analytic conductor: \(4.32261\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5475600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.149507714\)
\(L(\frac12)\) \(\approx\) \(4.149507714\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
13$C_2$ \( 1 - 2 T + p T^{2} \)
good7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.ab_ag
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.11.ad_ac
17$C_2^2$ \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.17.d_ai
19$C_2^2$ \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.19.f_g
23$C_2^2$ \( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.23.aj_cg
29$C_2^2$ \( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.29.j_ca
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.31.aq_ew
37$C_2^2$ \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.37.ah_m
41$C_2^2$ \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.41.ad_abg
43$C_2^2$ \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) 2.43.ab_abq
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.53.m_fm
59$C_2^2$ \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.59.aj_w
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.61.ab_aci
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.67.f_abq
71$C_2^2$ \( 1 - 9 T + 10 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.71.aj_k
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.73.ae_fu
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.79.aq_io
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2^2$ \( 1 - 3 T - 80 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.89.ad_adc
97$C_2^2$ \( 1 + 17 T + 192 T^{2} + 17 p T^{3} + p^{2} T^{4} \) 2.97.r_hk
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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

−9.211966508324690687834619285038, −8.815663704734760938512032144373, −8.553429383932978960344497285537, −8.031054992288858227986354698404, −7.77099735324449532980487119907, −7.13477955576465337576446470334, −6.66370491211076931508791306262, −6.57257019774244667581609677797, −6.04086070554811934187130170812, −5.86751479854887688953622662848, −5.09571113841103880455206903261, −4.90342054065998181266788109978, −4.21352412332947996950622474244, −4.21061932549473637190861834557, −3.35175041987842435503183080552, −2.96847211873991689843911186418, −2.15655547608844431942471640989, −2.13129367445823359700598163322, −1.15075681083104921035062135474, −0.811446194337417660079124394338, 0.811446194337417660079124394338, 1.15075681083104921035062135474, 2.13129367445823359700598163322, 2.15655547608844431942471640989, 2.96847211873991689843911186418, 3.35175041987842435503183080552, 4.21061932549473637190861834557, 4.21352412332947996950622474244, 4.90342054065998181266788109978, 5.09571113841103880455206903261, 5.86751479854887688953622662848, 6.04086070554811934187130170812, 6.57257019774244667581609677797, 6.66370491211076931508791306262, 7.13477955576465337576446470334, 7.77099735324449532980487119907, 8.031054992288858227986354698404, 8.553429383932978960344497285537, 8.815663704734760938512032144373, 9.211966508324690687834619285038

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