Properties

Label 4-191664-1.1-c1e2-0-8
Degree $4$
Conductor $191664$
Sign $1$
Analytic cond. $12.2206$
Root an. cond. $1.86970$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s − 4·5-s + 8·7-s − 3·8-s + 9-s − 4·10-s + 11-s + 8·14-s − 16-s + 18-s + 4·20-s + 22-s + 2·25-s − 8·28-s + 5·32-s − 32·35-s − 36-s + 12·37-s + 12·40-s − 44-s − 4·45-s + 34·49-s + 2·50-s + 12·53-s − 4·55-s − 24·56-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 1.78·5-s + 3.02·7-s − 1.06·8-s + 1/3·9-s − 1.26·10-s + 0.301·11-s + 2.13·14-s − 1/4·16-s + 0.235·18-s + 0.894·20-s + 0.213·22-s + 2/5·25-s − 1.51·28-s + 0.883·32-s − 5.40·35-s − 1/6·36-s + 1.97·37-s + 1.89·40-s − 0.150·44-s − 0.596·45-s + 34/7·49-s + 0.282·50-s + 1.64·53-s − 0.539·55-s − 3.20·56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 191664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 191664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(191664\)    =    \(2^{4} \cdot 3^{2} \cdot 11^{3}\)
Sign: $1$
Analytic conductor: \(12.2206\)
Root analytic conductor: \(1.86970\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 191664,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.020786204\)
\(L(\frac12)\) \(\approx\) \(2.020786204\)
\(L(\frac{3}{2})\) 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)$
bad2$C_2$ \( 1 - T + p T^{2} \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
11$C_1$ \( 1 - T \)
good5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
43$C_2$ \( ( 1 + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−8.975261543279013968212899152873, −8.388317550066773340937260013490, −8.021755838874073182736536348782, −7.890643100699516116601167538593, −7.46995816184461358823952634116, −6.80474099018451965837235050932, −5.99052260380604420685058602068, −5.31622482563829752699303340190, −5.03520265044937013447420220349, −4.30674031395560359136174367267, −4.20407903283329927801590362766, −3.84602405282169040782497156344, −2.78748407328752201898486204249, −1.85717409091206548414350568811, −0.889704343842193223159753487670, 0.889704343842193223159753487670, 1.85717409091206548414350568811, 2.78748407328752201898486204249, 3.84602405282169040782497156344, 4.20407903283329927801590362766, 4.30674031395560359136174367267, 5.03520265044937013447420220349, 5.31622482563829752699303340190, 5.99052260380604420685058602068, 6.80474099018451965837235050932, 7.46995816184461358823952634116, 7.890643100699516116601167538593, 8.021755838874073182736536348782, 8.388317550066773340937260013490, 8.975261543279013968212899152873

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