Properties

Label 2-312-1.1-c1-0-5
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 4·5-s − 4·7-s + 9-s − 2·11-s − 13-s − 4·15-s − 6·17-s + 4·19-s − 4·21-s + 4·23-s + 11·25-s + 27-s − 6·29-s + 8·31-s − 2·33-s + 16·35-s − 10·37-s − 39-s − 4·41-s − 4·43-s − 4·45-s − 6·47-s + 9·49-s − 6·51-s + 6·53-s + 8·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.78·5-s − 1.51·7-s + 1/3·9-s − 0.603·11-s − 0.277·13-s − 1.03·15-s − 1.45·17-s + 0.917·19-s − 0.872·21-s + 0.834·23-s + 11/5·25-s + 0.192·27-s − 1.11·29-s + 1.43·31-s − 0.348·33-s + 2.70·35-s − 1.64·37-s − 0.160·39-s − 0.624·41-s − 0.609·43-s − 0.596·45-s − 0.875·47-s + 9/7·49-s − 0.840·51-s + 0.824·53-s + 1.07·55-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: \(1\)
Selberg data: \((2,\ 312,\ (\ :1/2),\ -1)\)

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$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
13 \( 1 + T \)
good5 \( 1 + 4 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 10 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
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

−11.29980704017641679673210124244, −10.26107768864454919066328161222, −9.194400876497935718960111344835, −8.357409761893690301535182897713, −7.32957775547896201770982786561, −6.69348036415102394684996086591, −4.87226101842987421096655374451, −3.67480739515335372157705798375, −2.92519921810644207783878050101, 0, 2.92519921810644207783878050101, 3.67480739515335372157705798375, 4.87226101842987421096655374451, 6.69348036415102394684996086591, 7.32957775547896201770982786561, 8.357409761893690301535182897713, 9.194400876497935718960111344835, 10.26107768864454919066328161222, 11.29980704017641679673210124244

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