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Properties

Label 4-8624e2-1.1-c1e2-0-4
Degree $4$
Conductor $74373376$
Sign $1$
Analytic cond. $4742.11$
Root an. cond. $8.29837$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 9-s − 2·11-s + 2·13-s + 4·17-s − 8·19-s − 8·23-s − 8·25-s − 6·27-s + 2·29-s − 8·31-s − 4·33-s − 4·37-s + 4·39-s − 4·41-s + 16·43-s + 4·47-s + 8·51-s + 8·53-s − 16·57-s + 6·59-s + 22·61-s − 2·67-s − 16·69-s + 20·71-s + 4·73-s − 16·75-s + ⋯
L(s)  = 1  + 1.15·3-s − 1/3·9-s − 0.603·11-s + 0.554·13-s + 0.970·17-s − 1.83·19-s − 1.66·23-s − 8/5·25-s − 1.15·27-s + 0.371·29-s − 1.43·31-s − 0.696·33-s − 0.657·37-s + 0.640·39-s − 0.624·41-s + 2.43·43-s + 0.583·47-s + 1.12·51-s + 1.09·53-s − 2.11·57-s + 0.781·59-s + 2.81·61-s − 0.244·67-s − 1.92·69-s + 2.37·71-s + 0.468·73-s − 1.84·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74373376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74373376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(74373376\)    =    \(2^{8} \cdot 7^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(4742.11\)
Root analytic conductor: \(8.29837\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 74373376,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.245823278\)
\(L(\frac12)\) \(\approx\) \(2.245823278\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
11$C_1$ \( ( 1 + T )^{2} \)
good3$D_{4}$ \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.3.ac_f
5$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.5.a_i
13$D_{4}$ \( 1 - 2 T + 19 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.13.ac_t
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.17.ae_bm
19$D_{4}$ \( 1 + 8 T + 52 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.19.i_ca
23$D_{4}$ \( 1 + 8 T + 60 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.23.i_ci
29$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.29.ac_ch
31$D_{4}$ \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.31.i_bu
37$D_{4}$ \( 1 + 4 T + 28 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.37.e_bc
41$D_{4}$ \( 1 + 4 T + 68 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.41.e_cq
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.43.aq_fu
47$D_{4}$ \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.47.ae_dm
53$D_{4}$ \( 1 - 8 T + 120 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.53.ai_eq
59$D_{4}$ \( 1 - 6 T + 125 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.59.ag_ev
61$D_{4}$ \( 1 - 22 T + 235 T^{2} - 22 p T^{3} + p^{2} T^{4} \) 2.61.aw_jb
67$D_{4}$ \( 1 + 2 T + 85 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.67.c_dh
71$D_{4}$ \( 1 - 20 T + 224 T^{2} - 20 p T^{3} + p^{2} T^{4} \) 2.71.au_iq
73$D_{4}$ \( 1 - 4 T + 148 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.73.ae_fs
79$D_{4}$ \( 1 + 2 T + 109 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.79.c_ef
83$D_{4}$ \( 1 + 4 T + 98 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.83.e_du
89$D_{4}$ \( 1 - 24 T + 314 T^{2} - 24 p T^{3} + p^{2} T^{4} \) 2.89.ay_mc
97$D_{4}$ \( 1 - 26 T + 355 T^{2} - 26 p T^{3} + p^{2} T^{4} \) 2.97.aba_nr
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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

−7.984685916907702346841095873631, −7.73065640091985532299721706343, −7.28252216463144750559730115490, −7.21557792825137379895327732378, −6.36212405314112440706095947757, −6.26554259254879213121100767062, −5.92984732464335701416633897244, −5.56966206939268821343513742456, −5.20322444694246075737809028837, −4.91303227161931492959890328581, −4.10273578298198848765686338613, −3.90179990463991841946929649493, −3.61109909799597930746788294504, −3.57543358182465510441565210953, −2.64086360797039109863525206752, −2.35028366247862087940960895788, −2.20091586010092612134459616998, −1.83365561692902300189506176844, −0.944682176578822211001332908348, −0.34269503535730182617994507306, 0.34269503535730182617994507306, 0.944682176578822211001332908348, 1.83365561692902300189506176844, 2.20091586010092612134459616998, 2.35028366247862087940960895788, 2.64086360797039109863525206752, 3.57543358182465510441565210953, 3.61109909799597930746788294504, 3.90179990463991841946929649493, 4.10273578298198848765686338613, 4.91303227161931492959890328581, 5.20322444694246075737809028837, 5.56966206939268821343513742456, 5.92984732464335701416633897244, 6.26554259254879213121100767062, 6.36212405314112440706095947757, 7.21557792825137379895327732378, 7.28252216463144750559730115490, 7.73065640091985532299721706343, 7.984685916907702346841095873631

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