Properties

Label 4-968e2-1.1-c1e2-0-16
Degree $4$
Conductor $937024$
Sign $1$
Analytic cond. $59.7454$
Root an. cond. $2.78020$
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-s − 5-s − 7-s − 4·9-s + 5·13-s + 15-s − 3·17-s − 3·19-s + 21-s − 8·23-s − 8·25-s + 6·27-s + 29-s − 9·31-s + 35-s − 17·37-s − 5·39-s + 17·41-s − 12·43-s + 4·45-s − 47-s − 12·49-s + 3·51-s − 53-s + 3·57-s − 7·59-s − 3·61-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 0.377·7-s − 4/3·9-s + 1.38·13-s + 0.258·15-s − 0.727·17-s − 0.688·19-s + 0.218·21-s − 1.66·23-s − 8/5·25-s + 1.15·27-s + 0.185·29-s − 1.61·31-s + 0.169·35-s − 2.79·37-s − 0.800·39-s + 2.65·41-s − 1.82·43-s + 0.596·45-s − 0.145·47-s − 1.71·49-s + 0.420·51-s − 0.137·53-s + 0.397·57-s − 0.911·59-s − 0.384·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 937024 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 937024 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(937024\)    =    \(2^{6} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(59.7454\)
Root analytic conductor: \(2.78020\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 937024,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
11 \( 1 \)
good3$D_{4}$ \( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} \) 2.3.b_f
5$D_{4}$ \( 1 + T + 9 T^{2} + p T^{3} + p^{2} T^{4} \) 2.5.b_j
7$D_{4}$ \( 1 + T + 13 T^{2} + p T^{3} + p^{2} T^{4} \) 2.7.b_n
13$D_{4}$ \( 1 - 5 T + 21 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.13.af_v
17$D_{4}$ \( 1 + 3 T + 25 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.17.d_z
19$D_{4}$ \( 1 + 3 T + 29 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.19.d_bd
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.23.i_ck
29$D_{4}$ \( 1 - T - 3 T^{2} - p T^{3} + p^{2} T^{4} \) 2.29.ab_ad
31$D_{4}$ \( 1 + 9 T + 51 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.31.j_bz
37$D_{4}$ \( 1 + 17 T + 145 T^{2} + 17 p T^{3} + p^{2} T^{4} \) 2.37.r_fp
41$D_{4}$ \( 1 - 17 T + 153 T^{2} - 17 p T^{3} + p^{2} T^{4} \) 2.41.ar_fx
43$D_{4}$ \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.43.m_dy
47$D_{4}$ \( 1 + T - 7 T^{2} + p T^{3} + p^{2} T^{4} \) 2.47.b_ah
53$D_{4}$ \( 1 + T + 105 T^{2} + p T^{3} + p^{2} T^{4} \) 2.53.b_eb
59$D_{4}$ \( 1 + 7 T + 129 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.59.h_ez
61$D_{4}$ \( 1 + 3 T + 113 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.61.d_ej
67$D_{4}$ \( 1 + 20 T + 214 T^{2} + 20 p T^{3} + p^{2} T^{4} \) 2.67.u_ig
71$D_{4}$ \( 1 - T + 111 T^{2} - p T^{3} + p^{2} T^{4} \) 2.71.ab_eh
73$D_{4}$ \( 1 - 13 T + 177 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.73.an_gv
79$D_{4}$ \( 1 - T + 147 T^{2} - p T^{3} + p^{2} T^{4} \) 2.79.ab_fr
83$D_{4}$ \( 1 + 21 T + 275 T^{2} + 21 p T^{3} + p^{2} T^{4} \) 2.83.v_kp
89$C_2^2$ \( 1 + 158 T^{2} + p^{2} T^{4} \) 2.89.a_gc
97$D_{4}$ \( 1 + 9 T + 153 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.97.j_fx
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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

−9.718155373631337909562928826012, −9.433779579397515233092921468592, −8.790340375267244736428367243560, −8.677796205558392982059395728661, −8.021564041432548748689502364324, −7.993669350098141213010103018783, −7.21955685717517008503985414319, −6.72500884991025223316708225768, −6.18786655007509705655446190507, −6.02617977199967583271809473827, −5.59912022531615973356578863948, −5.15273988382638544398513003784, −4.23320561779896701245064714031, −4.13992629092158994388046226367, −3.28203777656175467468818893269, −3.19015806752990488848030640151, −2.02723977633033867325150241230, −1.70035795728857034723097312919, 0, 0, 1.70035795728857034723097312919, 2.02723977633033867325150241230, 3.19015806752990488848030640151, 3.28203777656175467468818893269, 4.13992629092158994388046226367, 4.23320561779896701245064714031, 5.15273988382638544398513003784, 5.59912022531615973356578863948, 6.02617977199967583271809473827, 6.18786655007509705655446190507, 6.72500884991025223316708225768, 7.21955685717517008503985414319, 7.993669350098141213010103018783, 8.021564041432548748689502364324, 8.677796205558392982059395728661, 8.790340375267244736428367243560, 9.433779579397515233092921468592, 9.718155373631337909562928826012

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