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Properties

Label 2-279-31.30-c0-0-2
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  + 2-s + 5-s − 7-s − 8-s + 10-s − 14-s − 16-s − 19-s + 31-s − 35-s − 38-s − 40-s + 41-s − 2·47-s + 56-s + 59-s + 62-s + 64-s + 2·67-s − 70-s + 71-s − 80-s + 82-s − 2·94-s − 95-s − 97-s + 101-s + ⋯
L(s)  = 1  + 2-s + 5-s − 7-s − 8-s + 10-s − 14-s − 16-s − 19-s + 31-s − 35-s − 38-s − 40-s + 41-s − 2·47-s + 56-s + 59-s + 62-s + 64-s + 2·67-s − 70-s + 71-s − 80-s + 82-s − 2·94-s − 95-s − 97-s + 101-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

\(L(\frac{1}{2})\) \(\approx\) \(1.105897203\)
\(L(\frac12)\) \(\approx\) \(1.105897203\)
\(L(1)\) 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 \( 1 - T + T^{2} \)
5 \( 1 - T + T^{2} \)
7 \( 1 + T + T^{2} \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( 1 + T + T^{2} \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( 1 - T + T^{2} \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( ( 1 + T )^{2} \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( 1 - T + T^{2} \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 - T )^{2} \)
71 \( 1 - T + T^{2} \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( 1 + T + T^{2} \)
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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

−12.56327422383395901883796235506, −11.35047144455912251746066164418, −10.02678969465852995637119360071, −9.477020025142260729168934848643, −8.379672981525320787108347947844, −6.61896191061210970729688741254, −6.07232293912711115418603469203, −4.99069585757982579655743675468, −3.75019253467280126294847026101, −2.50339585630080990501102233300, 2.50339585630080990501102233300, 3.75019253467280126294847026101, 4.99069585757982579655743675468, 6.07232293912711115418603469203, 6.61896191061210970729688741254, 8.379672981525320787108347947844, 9.477020025142260729168934848643, 10.02678969465852995637119360071, 11.35047144455912251746066164418, 12.56327422383395901883796235506

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