Properties

Label 2-312-1.1-c1-0-1
Degree $2$
Conductor $312$
Sign $1$
Analytic cond. $2.49133$
Root an. cond. $1.57839$
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  − 3-s + 4·5-s + 9-s − 2·11-s − 13-s − 4·15-s + 2·17-s + 8·19-s + 4·23-s + 11·25-s − 27-s − 6·29-s − 4·31-s + 2·33-s + 6·37-s + 39-s − 12·41-s + 4·43-s + 4·45-s − 6·47-s − 7·49-s − 2·51-s − 2·53-s − 8·55-s − 8·57-s − 14·59-s + 10·61-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.78·5-s + 1/3·9-s − 0.603·11-s − 0.277·13-s − 1.03·15-s + 0.485·17-s + 1.83·19-s + 0.834·23-s + 11/5·25-s − 0.192·27-s − 1.11·29-s − 0.718·31-s + 0.348·33-s + 0.986·37-s + 0.160·39-s − 1.87·41-s + 0.609·43-s + 0.596·45-s − 0.875·47-s − 49-s − 0.280·51-s − 0.274·53-s − 1.07·55-s − 1.05·57-s − 1.82·59-s + 1.28·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $1$
Analytic conductor: \(2.49133\)
Root analytic conductor: \(1.57839\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{312} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 312,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.441654840\)
\(L(\frac12)\) \(\approx\) \(1.441654840\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
13 \( 1 + T \)
good5 \( 1 - 4 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.59357135369944919222989323737, −10.63466219619293519478353011311, −9.764486816476355431308846737996, −9.281879014062647170255646811904, −7.69837808741404231925046977179, −6.63559690348924214052369009469, −5.53392832912108757065282436695, −5.11677941303251687475460824152, −3.02936751855165763048811417152, −1.54183448393116236574503713957, 1.54183448393116236574503713957, 3.02936751855165763048811417152, 5.11677941303251687475460824152, 5.53392832912108757065282436695, 6.63559690348924214052369009469, 7.69837808741404231925046977179, 9.281879014062647170255646811904, 9.764486816476355431308846737996, 10.63466219619293519478353011311, 11.59357135369944919222989323737

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