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Properties

Label 4-7488e2-1.1-c1e2-0-10
Degree $4$
Conductor $56070144$
Sign $1$
Analytic cond. $3575.08$
Root an. cond. $7.73252$
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  − 3·5-s − 3·7-s − 4·11-s + 2·13-s − 3·17-s + 12·19-s + 25-s − 6·31-s + 9·35-s + 7·37-s − 6·41-s + 3·43-s − 9·47-s − 3·49-s − 18·53-s + 12·55-s + 12·59-s − 2·61-s − 6·65-s + 12·67-s + 9·71-s + 20·73-s + 12·77-s − 24·79-s + 10·83-s + 9·85-s − 6·91-s + ⋯
L(s)  = 1  − 1.34·5-s − 1.13·7-s − 1.20·11-s + 0.554·13-s − 0.727·17-s + 2.75·19-s + 1/5·25-s − 1.07·31-s + 1.52·35-s + 1.15·37-s − 0.937·41-s + 0.457·43-s − 1.31·47-s − 3/7·49-s − 2.47·53-s + 1.61·55-s + 1.56·59-s − 0.256·61-s − 0.744·65-s + 1.46·67-s + 1.06·71-s + 2.34·73-s + 1.36·77-s − 2.70·79-s + 1.09·83-s + 0.976·85-s − 0.628·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56070144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56070144 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(56070144\)    =    \(2^{12} \cdot 3^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(3575.08\)
Root analytic conductor: \(7.73252\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 56070144,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
13$C_1$ \( ( 1 - T )^{2} \)
good5$C_2^2$ \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.5.d_i
7$D_{4}$ \( 1 + 3 T + 12 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.7.d_m
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.11.e_ba
17$D_{4}$ \( 1 + 3 T + 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.17.d_bg
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.19.am_cw
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.29.a_ak
31$D_{4}$ \( 1 + 6 T + 54 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.31.g_cc
37$D_{4}$ \( 1 - 7 T + 48 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.37.ah_bw
41$D_{4}$ \( 1 + 6 T + 74 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.41.g_cw
43$D_{4}$ \( 1 - 3 T - 18 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.43.ad_as
47$D_{4}$ \( 1 + 9 T + 76 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.47.j_cy
53$D_{4}$ \( 1 + 18 T + 170 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.53.s_go
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.59.am_fy
61$D_{4}$ \( 1 + 2 T - 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.61.c_abe
67$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.67.am_go
71$D_{4}$ \( 1 - 9 T + 124 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.71.aj_eu
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.73.au_jm
79$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.79.y_lq
83$D_{4}$ \( 1 - 10 T + 38 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.83.ak_bm
89$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \) 2.89.a_eg
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.97.m_iw
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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

−7.79308363445361596276870766287, −7.55228482028089024145500858943, −6.89617490133496935782742054236, −6.85727833356832965090982119725, −6.39278736114962877857692711778, −6.04447702844475662948161824141, −5.41709986110003456164548110451, −5.35731882932171156221690425150, −4.89220322364161869229626293289, −4.59628792190928086812314168034, −3.81718100495748970790071769574, −3.79139281928885828607401100705, −3.22877402474230662825821815318, −3.21752555051964748584457874657, −2.59075488315169351688316389860, −2.16358800842261393298224165112, −1.38343299650664468285594617039, −0.913949618391616138541189427665, 0, 0, 0.913949618391616138541189427665, 1.38343299650664468285594617039, 2.16358800842261393298224165112, 2.59075488315169351688316389860, 3.21752555051964748584457874657, 3.22877402474230662825821815318, 3.79139281928885828607401100705, 3.81718100495748970790071769574, 4.59628792190928086812314168034, 4.89220322364161869229626293289, 5.35731882932171156221690425150, 5.41709986110003456164548110451, 6.04447702844475662948161824141, 6.39278736114962877857692711778, 6.85727833356832965090982119725, 6.89617490133496935782742054236, 7.55228482028089024145500858943, 7.79308363445361596276870766287

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