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Properties

Label 4-2142e2-1.1-c1e2-0-11
Degree $4$
Conductor $4588164$
Sign $1$
Analytic cond. $292.545$
Root an. cond. $4.13569$
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  + 2·2-s + 3·4-s − 2·5-s − 2·7-s + 4·8-s − 4·10-s − 6·11-s + 4·13-s − 4·14-s + 5·16-s − 2·17-s − 8·19-s − 6·20-s − 12·22-s − 16·23-s − 2·25-s + 8·26-s − 6·28-s − 2·29-s − 12·31-s + 6·32-s − 4·34-s + 4·35-s − 2·37-s − 16·38-s − 8·40-s + 16·41-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s − 0.894·5-s − 0.755·7-s + 1.41·8-s − 1.26·10-s − 1.80·11-s + 1.10·13-s − 1.06·14-s + 5/4·16-s − 0.485·17-s − 1.83·19-s − 1.34·20-s − 2.55·22-s − 3.33·23-s − 2/5·25-s + 1.56·26-s − 1.13·28-s − 0.371·29-s − 2.15·31-s + 1.06·32-s − 0.685·34-s + 0.676·35-s − 0.328·37-s − 2.59·38-s − 1.26·40-s + 2.49·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4588164 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4588164 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4588164\)    =    \(2^{2} \cdot 3^{4} \cdot 7^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(292.545\)
Root analytic conductor: \(4.13569\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 4588164,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 - T )^{2} \)
3 \( 1 \)
7$C_1$ \( ( 1 + T )^{2} \)
17$C_1$ \( ( 1 + T )^{2} \)
good5$D_{4}$ \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.5.c_g
11$C_4$ \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.11.g_ba
13$D_{4}$ \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.13.ae_k
19$C_4$ \( 1 + 8 T + 34 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.19.i_bi
23$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.23.q_eg
29$D_{4}$ \( 1 + 2 T + 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.29.c_o
31$D_{4}$ \( 1 + 12 T + 78 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.31.m_da
37$D_{4}$ \( 1 + 2 T + 70 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.37.c_cs
41$C_4$ \( 1 - 16 T + 126 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.41.aq_ew
43$D_{4}$ \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.43.e_cs
47$D_{4}$ \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.47.e_da
53$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \) 2.53.a_di
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.59.am_fy
61$D_{4}$ \( 1 + 2 T + 118 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.61.c_eo
67$D_{4}$ \( 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.67.m_fu
71$C_4$ \( 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.71.e_ew
73$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.73.a_abi
79$D_{4}$ \( 1 + 12 T + 174 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.79.m_gs
83$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.83.e_go
89$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.89.e_ha
97$D_{4}$ \( 1 + 8 T + 190 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.97.i_hi
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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

−8.530487579100839530958061451392, −8.519643170441479184196798350180, −7.906608943505178757293760547220, −7.79876486141602146032934699339, −7.22116646065203033464362963348, −6.98309760699053505568548655819, −6.16524167532584390439964660991, −6.14553919480875602965460249142, −5.62455075351154074304458195750, −5.57243703951730312572015805950, −4.65859373517202542072828387902, −4.28795130737825788549991806637, −3.86630398631097085718473349476, −3.84331465544391044260854174398, −3.17318402346990975243291590030, −2.52251175298772374350130819622, −2.15968805105653392566630340750, −1.69977336870992701193419165366, 0, 0, 1.69977336870992701193419165366, 2.15968805105653392566630340750, 2.52251175298772374350130819622, 3.17318402346990975243291590030, 3.84331465544391044260854174398, 3.86630398631097085718473349476, 4.28795130737825788549991806637, 4.65859373517202542072828387902, 5.57243703951730312572015805950, 5.62455075351154074304458195750, 6.14553919480875602965460249142, 6.16524167532584390439964660991, 6.98309760699053505568548655819, 7.22116646065203033464362963348, 7.79876486141602146032934699339, 7.906608943505178757293760547220, 8.519643170441479184196798350180, 8.530487579100839530958061451392

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