Properties

Label 4-100312-1.1-c1e2-0-0
Degree $4$
Conductor $100312$
Sign $1$
Analytic cond. $6.39598$
Root an. cond. $1.59029$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 6-s − 4·7-s + 8-s − 9-s + 6·11-s + 12-s − 4·13-s − 4·14-s + 16-s + 6·17-s − 18-s + 3·19-s − 4·21-s + 6·22-s + 24-s + 2·25-s − 4·26-s − 4·28-s − 2·29-s − 31-s + 32-s + 6·33-s + 6·34-s − 36-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.51·7-s + 0.353·8-s − 1/3·9-s + 1.80·11-s + 0.288·12-s − 1.10·13-s − 1.06·14-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.688·19-s − 0.872·21-s + 1.27·22-s + 0.204·24-s + 2/5·25-s − 0.784·26-s − 0.755·28-s − 0.371·29-s − 0.179·31-s + 0.176·32-s + 1.04·33-s + 1.02·34-s − 1/6·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100312 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100312 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(100312\)    =    \(2^{3} \cdot 12539\)
Sign: $1$
Analytic conductor: \(6.39598\)
Root analytic conductor: \(1.59029\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 100312,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.649361853\)
\(L(\frac12)\) \(\approx\) \(2.649361853\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
12539$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 40 T + p T^{2} ) \)
good3$D_{4}$ \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
13$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
19$C_2^2$ \( 1 - 3 T + 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 28 T^{2} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + T - 38 T^{2} + p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 7 T + 52 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
41$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$D_{4}$ \( 1 - 6 T + 88 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 4 T - 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 6 T + 110 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 7 T + 52 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 88 T^{2} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 2 T + 52 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 6 T + 2 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 11 T + 170 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 5 T + 110 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 3 T + 172 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 4 T + 126 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
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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

−14.2015223461, −13.7047089339, −13.2283787441, −12.6365480801, −12.3406530300, −12.1184906079, −11.5473537611, −11.1146903724, −10.4237047227, −9.78713444493, −9.61319157347, −9.25487314618, −8.71036294143, −7.96904059350, −7.44747705984, −7.04225593867, −6.46398421969, −6.03891012539, −5.51452862274, −4.79805001258, −4.00274497500, −3.54872831445, −3.02578496435, −2.43820375580, −1.13047116433, 1.13047116433, 2.43820375580, 3.02578496435, 3.54872831445, 4.00274497500, 4.79805001258, 5.51452862274, 6.03891012539, 6.46398421969, 7.04225593867, 7.44747705984, 7.96904059350, 8.71036294143, 9.25487314618, 9.61319157347, 9.78713444493, 10.4237047227, 11.1146903724, 11.5473537611, 12.1184906079, 12.3406530300, 12.6365480801, 13.2283787441, 13.7047089339, 14.2015223461

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