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Properties

Label 4-18976-1.1-c1e2-0-0
Degree $4$
Conductor $18976$
Sign $-1$
Analytic cond. $1.20992$
Root an. cond. $1.04879$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 6·5-s + 8-s − 2·9-s − 6·10-s − 10·13-s + 16-s + 4·17-s − 2·18-s − 6·20-s + 18·25-s − 10·26-s − 4·29-s + 32-s + 4·34-s − 2·36-s + 4·37-s − 6·40-s − 4·41-s + 12·45-s − 6·49-s + 18·50-s − 10·52-s + 8·53-s − 4·58-s − 10·61-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 2.68·5-s + 0.353·8-s − 2/3·9-s − 1.89·10-s − 2.77·13-s + 1/4·16-s + 0.970·17-s − 0.471·18-s − 1.34·20-s + 18/5·25-s − 1.96·26-s − 0.742·29-s + 0.176·32-s + 0.685·34-s − 1/3·36-s + 0.657·37-s − 0.948·40-s − 0.624·41-s + 1.78·45-s − 6/7·49-s + 2.54·50-s − 1.38·52-s + 1.09·53-s − 0.525·58-s − 1.28·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 18976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18976 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(18976\)    =    \(2^{5} \cdot 593\)
Sign: $-1$
Analytic conductor: \(1.20992\)
Root analytic conductor: \(1.04879\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 18976,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 - T \)
593$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 14 T + p T^{2} ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.3.a_c
5$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.g_s
7$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.7.a_g
11$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.11.a_c
13$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.k_by
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.ae_w
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.19.a_k
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.29.e_cg
31$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.31.a_ag
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.37.ae_o
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.e_w
43$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.43.a_bi
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.47.a_acg
53$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.53.ai_cg
59$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \) 2.59.a_abm
61$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.61.k_fq
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.67.a_k
71$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.71.a_ade
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.73.e_fu
79$C_2^2$ \( 1 + 106 T^{2} + p^{2} T^{4} \) 2.79.a_ec
83$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.83.a_bu
89$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.89.aq_fm
97$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.e_cc
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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

−10.90394425068887150729289423901, −10.21558741160790470317225741485, −9.670551344995884280691450922121, −8.899146401148202841220955403631, −8.113788572517057482945092416867, −7.74872044538728673688316898772, −7.38120798313998015865955982486, −7.01123405494208093760238226250, −5.94530251173618386187442757523, −5.02431750313193489962863289746, −4.72071085290983858899319297867, −3.92687056366562963793925901966, −3.31781764934721215450415253059, −2.56986036889549924612058123858, 0, 2.56986036889549924612058123858, 3.31781764934721215450415253059, 3.92687056366562963793925901966, 4.72071085290983858899319297867, 5.02431750313193489962863289746, 5.94530251173618386187442757523, 7.01123405494208093760238226250, 7.38120798313998015865955982486, 7.74872044538728673688316898772, 8.113788572517057482945092416867, 8.899146401148202841220955403631, 9.670551344995884280691450922121, 10.21558741160790470317225741485, 10.90394425068887150729289423901

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