Properties

Label 2-279-31.30-c0-0-1
Degree $2$
Conductor $279$
Sign $1$
Analytic cond. $0.139239$
Root an. cond. $0.373147$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s + 2·7-s + 16-s − 2·19-s − 25-s − 2·28-s − 31-s + 3·49-s − 64-s − 2·67-s + 2·76-s + 2·97-s + 100-s − 2·103-s + 2·109-s + 2·112-s + ⋯
L(s)  = 1  − 4-s + 2·7-s + 16-s − 2·19-s − 25-s − 2·28-s − 31-s + 3·49-s − 64-s − 2·67-s + 2·76-s + 2·97-s + 100-s − 2·103-s + 2·109-s + 2·112-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 279 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 279 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(279\)    =    \(3^{2} \cdot 31\)
Sign: $1$
Analytic conductor: \(0.139239\)
Root analytic conductor: \(0.373147\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{279} (154, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 279,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7216711995\)
\(L(\frac12)\) \(\approx\) \(0.7216711995\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 + T \)
good2 \( 1 + T^{2} \)
5 \( 1 + T^{2} \)
7 \( ( 1 - T )^{2} \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( ( 1 + T )^{2} \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( 1 + T^{2} \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( 1 + T^{2} \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( 1 + T^{2} \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 + T )^{2} \)
71 \( 1 + T^{2} \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( ( 1 - T )^{2} \)
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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

−12.08018341524497221045736657749, −11.09521140809559409479802410105, −10.33599727177141278791702208469, −8.992590991605716742617066459935, −8.348448212594379224261787842844, −7.54688362034994282287623546266, −5.85143602856598050045515693047, −4.79360867515447630427111469638, −4.06366238061491538956308440142, −1.86486858667779050269360380283, 1.86486858667779050269360380283, 4.06366238061491538956308440142, 4.79360867515447630427111469638, 5.85143602856598050045515693047, 7.54688362034994282287623546266, 8.348448212594379224261787842844, 8.992590991605716742617066459935, 10.33599727177141278791702208469, 11.09521140809559409479802410105, 12.08018341524497221045736657749

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