Properties

Label 4-1656e2-1.1-c1e2-0-5
Degree $4$
Conductor $2742336$
Sign $1$
Analytic cond. $174.853$
Root an. cond. $3.63637$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 2·7-s − 4·11-s − 4·13-s − 2·17-s − 6·19-s − 2·23-s − 4·25-s + 4·29-s − 12·31-s − 4·35-s − 8·37-s − 2·43-s − 8·47-s − 8·49-s + 18·53-s − 8·55-s + 8·59-s − 8·61-s − 8·65-s − 2·67-s − 8·71-s − 12·73-s + 8·77-s + 6·79-s − 12·83-s − 4·85-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.755·7-s − 1.20·11-s − 1.10·13-s − 0.485·17-s − 1.37·19-s − 0.417·23-s − 4/5·25-s + 0.742·29-s − 2.15·31-s − 0.676·35-s − 1.31·37-s − 0.304·43-s − 1.16·47-s − 8/7·49-s + 2.47·53-s − 1.07·55-s + 1.04·59-s − 1.02·61-s − 0.992·65-s − 0.244·67-s − 0.949·71-s − 1.40·73-s + 0.911·77-s + 0.675·79-s − 1.31·83-s − 0.433·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2742336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2742336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2742336\)    =    \(2^{6} \cdot 3^{4} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(174.853\)
Root analytic conductor: \(3.63637\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 2742336,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
23$C_1$ \( ( 1 + T )^{2} \)
good5$D_{4}$ \( 1 - 2 T + 8 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.5.ac_i
7$D_{4}$ \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_m
11$D_{4}$ \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.11.e_o
13$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.13.e_s
17$D_{4}$ \( 1 + 2 T + 32 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.17.c_bg
19$D_{4}$ \( 1 + 6 T + 44 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.19.g_bs
29$D_{4}$ \( 1 - 4 T + 50 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.29.ae_by
31$D_{4}$ \( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.31.m_di
37$D_{4}$ \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.37.i_da
41$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.41.a_bi
43$D_{4}$ \( 1 + 2 T + 84 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.43.c_dg
47$D_{4}$ \( 1 + 8 T + 98 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.47.i_du
53$D_{4}$ \( 1 - 18 T + 160 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.53.as_ge
59$D_{4}$ \( 1 - 8 T + 122 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.59.ai_es
61$D_{4}$ \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.61.i_be
67$D_{4}$ \( 1 + 2 T + 108 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.67.c_ee
71$D_{4}$ \( 1 + 8 T + 110 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.71.i_eg
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.73.m_ha
79$D_{4}$ \( 1 - 6 T + 20 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.79.ag_u
83$D_{4}$ \( 1 + 12 T + 94 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.83.m_dq
89$D_{4}$ \( 1 + 2 T + 152 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.89.c_fw
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.97.e_hq
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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

−9.200412461894217812181139664143, −8.866794707661028322275496497763, −8.254433695245695620088875480956, −8.213777730606585566556791802700, −7.36073639052160638109740602630, −7.27945996953248953744877223326, −6.69176280745611718320861006903, −6.44042636429945810186621779230, −5.71826703275029143133431763030, −5.66513851201071782121094883552, −5.10242795554749919342498801270, −4.72692827565538444668490835445, −4.01895384090146151565072790948, −3.75548158582844565455365194059, −2.81512187685138659253017321130, −2.71617276138819515091728098689, −1.90601417390600743823822962909, −1.74732400062900900786872724625, 0, 0, 1.74732400062900900786872724625, 1.90601417390600743823822962909, 2.71617276138819515091728098689, 2.81512187685138659253017321130, 3.75548158582844565455365194059, 4.01895384090146151565072790948, 4.72692827565538444668490835445, 5.10242795554749919342498801270, 5.66513851201071782121094883552, 5.71826703275029143133431763030, 6.44042636429945810186621779230, 6.69176280745611718320861006903, 7.27945996953248953744877223326, 7.36073639052160638109740602630, 8.213777730606585566556791802700, 8.254433695245695620088875480956, 8.866794707661028322275496497763, 9.200412461894217812181139664143

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