Properties

Label 4-1656e2-1.1-c1e2-0-4
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  − 4·5-s − 2·11-s + 5·13-s − 2·17-s + 2·19-s + 2·23-s + 2·25-s − 3·29-s − 9·31-s − 41-s − 16·43-s − 11·47-s − 14·49-s − 4·53-s + 8·55-s − 4·59-s + 8·61-s − 20·65-s − 2·67-s − 23·71-s − 17·73-s − 2·79-s − 12·83-s + 8·85-s + 2·89-s − 8·95-s − 2·97-s + ⋯
L(s)  = 1  − 1.78·5-s − 0.603·11-s + 1.38·13-s − 0.485·17-s + 0.458·19-s + 0.417·23-s + 2/5·25-s − 0.557·29-s − 1.61·31-s − 0.156·41-s − 2.43·43-s − 1.60·47-s − 2·49-s − 0.549·53-s + 1.07·55-s − 0.520·59-s + 1.02·61-s − 2.48·65-s − 0.244·67-s − 2.72·71-s − 1.98·73-s − 0.225·79-s − 1.31·83-s + 0.867·85-s + 0.211·89-s − 0.820·95-s − 0.203·97-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$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.5.e_o
7$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.7.a_o
11$D_{4}$ \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.11.c_g
13$D_{4}$ \( 1 - 5 T + 28 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.13.af_bc
17$D_{4}$ \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.17.c_s
19$D_{4}$ \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.19.ac_w
29$D_{4}$ \( 1 + 3 T + 56 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.29.d_ce
31$D_{4}$ \( 1 + 9 T + 78 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.31.j_da
37$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.37.a_g
41$D_{4}$ \( 1 + T - 24 T^{2} + p T^{3} + p^{2} T^{4} \) 2.41.b_ay
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.43.q_fu
47$D_{4}$ \( 1 + 11 T + 86 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.47.l_di
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.53.e_eg
59$D_{4}$ \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.59.e_cc
61$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.61.ai_cs
67$D_{4}$ \( 1 + 2 T + 118 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.67.c_eo
71$D_{4}$ \( 1 + 23 T + 270 T^{2} + 23 p T^{3} + p^{2} T^{4} \) 2.71.x_kk
73$D_{4}$ \( 1 + 17 T + 180 T^{2} + 17 p T^{3} + p^{2} T^{4} \) 2.73.r_gy
79$D_{4}$ \( 1 + 2 T + 142 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.79.c_fm
83$D_{4}$ \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.83.m_fe
89$D_{4}$ \( 1 - 2 T + 26 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.89.ac_ba
97$D_{4}$ \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.97.c_bq
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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

−8.976249067349437709764688785485, −8.746736385064706737840619030030, −8.186422433254707433153706269900, −8.124290424184138514192108824047, −7.55348928553054491057742265065, −7.40591581544749298185377153402, −6.67963513332516710599777904649, −6.59877885594318682548851978249, −5.85165657463478403100488548936, −5.48826647517209078756682284807, −5.00415181104012127363492075101, −4.45959795138111280343625868226, −4.10806812399186429025010777766, −3.59820751860751599702084420667, −3.20717455827987912561325873685, −2.95805025675360105325132450644, −1.64074797369415812029192485669, −1.59770949758107538041436817562, 0, 0, 1.59770949758107538041436817562, 1.64074797369415812029192485669, 2.95805025675360105325132450644, 3.20717455827987912561325873685, 3.59820751860751599702084420667, 4.10806812399186429025010777766, 4.45959795138111280343625868226, 5.00415181104012127363492075101, 5.48826647517209078756682284807, 5.85165657463478403100488548936, 6.59877885594318682548851978249, 6.67963513332516710599777904649, 7.40591581544749298185377153402, 7.55348928553054491057742265065, 8.124290424184138514192108824047, 8.186422433254707433153706269900, 8.746736385064706737840619030030, 8.976249067349437709764688785485

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