Properties

Label 4-93312-1.1-c1e2-0-5
Degree $4$
Conductor $93312$
Sign $1$
Analytic cond. $5.94965$
Root an. cond. $1.56179$
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 − 2·7-s + 8-s − 2·14-s + 16-s + 12·23-s − 25-s − 2·28-s + 10·31-s + 32-s + 12·41-s + 12·46-s − 12·47-s − 11·49-s − 50-s − 2·56-s + 10·62-s + 64-s − 14·73-s + 16·79-s + 12·82-s + 36·89-s + 12·92-s − 12·94-s − 2·97-s − 11·98-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.755·7-s + 0.353·8-s − 0.534·14-s + 1/4·16-s + 2.50·23-s − 1/5·25-s − 0.377·28-s + 1.79·31-s + 0.176·32-s + 1.87·41-s + 1.76·46-s − 1.75·47-s − 1.57·49-s − 0.141·50-s − 0.267·56-s + 1.27·62-s + 1/8·64-s − 1.63·73-s + 1.80·79-s + 1.32·82-s + 3.81·89-s + 1.25·92-s − 1.23·94-s − 0.203·97-s − 1.11·98-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: $\chi_{93312} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 93312,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.252789771\)
\(L(\frac12)\) \(\approx\) \(2.252789771\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
3 \( 1 \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
7$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
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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

−9.512741326522107701681952133776, −9.254274842104821367212382551306, −8.728358107276488427256715933065, −7.82723843094092867969941089213, −7.78966227402229455461895642068, −6.71508725988629390136097161445, −6.69450931342145073952637354839, −6.11240325026052414300507082707, −5.39588744112665066773671436635, −4.77709333112371596512170551343, −4.45861885513123914392518249053, −3.38549178369413487864105189864, −3.13246179353932073023232514138, −2.34838622997525881980959331572, −1.07855163027675466308171882838, 1.07855163027675466308171882838, 2.34838622997525881980959331572, 3.13246179353932073023232514138, 3.38549178369413487864105189864, 4.45861885513123914392518249053, 4.77709333112371596512170551343, 5.39588744112665066773671436635, 6.11240325026052414300507082707, 6.69450931342145073952637354839, 6.71508725988629390136097161445, 7.78966227402229455461895642068, 7.82723843094092867969941089213, 8.728358107276488427256715933065, 9.254274842104821367212382551306, 9.512741326522107701681952133776

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