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Properties

Label 4-93312-1.1-c1e2-0-30
Degree $4$
Conductor $93312$
Sign $-1$
Analytic cond. $5.94965$
Root an. cond. $1.56179$
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 + 3-s + 4-s − 6-s − 8-s + 9-s − 6·11-s + 12-s + 16-s − 9·17-s − 18-s + 19-s + 6·22-s − 24-s + 2·25-s + 27-s − 32-s − 6·33-s + 9·34-s + 36-s − 38-s − 3·41-s − 2·43-s − 6·44-s + 48-s − 4·49-s − 2·50-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 1.80·11-s + 0.288·12-s + 1/4·16-s − 2.18·17-s − 0.235·18-s + 0.229·19-s + 1.27·22-s − 0.204·24-s + 2/5·25-s + 0.192·27-s − 0.176·32-s − 1.04·33-s + 1.54·34-s + 1/6·36-s − 0.162·38-s − 0.468·41-s − 0.304·43-s − 0.904·44-s + 0.144·48-s − 4/7·49-s − 0.282·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(93312\)    =    \(2^{7} \cdot 3^{6}\)
Sign: $-1$
Analytic conductor: \(5.94965\)
Root analytic conductor: \(1.56179\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 93312,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 + T \)
3$C_1$ \( 1 - T \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
7$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.7.a_e
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.11.g_bf
13$C_2^2$ \( 1 + 19 T^{2} + p^{2} T^{4} \) 2.13.a_t
17$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.j_ca
19$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.ab_s
23$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.23.a_ac
29$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.29.a_ai
31$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.31.a_aba
37$C_2^2$ \( 1 + 61 T^{2} + p^{2} T^{4} \) 2.37.a_cj
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.41.d_bc
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.43.c_g
47$C_2^2$ \( 1 - 89 T^{2} + p^{2} T^{4} \) 2.47.a_adl
53$C_2^2$ \( 1 - 32 T^{2} + p^{2} T^{4} \) 2.53.a_abg
59$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.59.ag_eo
61$C_2^2$ \( 1 + 85 T^{2} + p^{2} T^{4} \) 2.61.a_dh
67$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.67.l_ee
71$C_2^2$ \( 1 + 103 T^{2} + p^{2} T^{4} \) 2.71.a_dz
73$C_2$$\times$$C_2$ \( ( 1 + 10 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.73.ba_lu
79$C_2^2$ \( 1 - 32 T^{2} + p^{2} T^{4} \) 2.79.a_abg
83$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.ad_ao
89$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.89.aj_fm
97$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.97.ae_en
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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

−9.238097176676544541856919447609, −8.755320776572624605657268556796, −8.589031677461357649762066369538, −7.86660308641488423632513423103, −7.55904839014797635084846447901, −6.97991898123414951150681433901, −6.49242845094790160210673561057, −5.85346497436345917136603741245, −5.07562986357202465204427362014, −4.66813422565207745216710594028, −3.87488613206077017862725546029, −2.89165547258907086481720434289, −2.55542664503741642179895212946, −1.71012766301984892032160078976, 0, 1.71012766301984892032160078976, 2.55542664503741642179895212946, 2.89165547258907086481720434289, 3.87488613206077017862725546029, 4.66813422565207745216710594028, 5.07562986357202465204427362014, 5.85346497436345917136603741245, 6.49242845094790160210673561057, 6.97991898123414951150681433901, 7.55904839014797635084846447901, 7.86660308641488423632513423103, 8.589031677461357649762066369538, 8.755320776572624605657268556796, 9.238097176676544541856919447609

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