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Properties

Label 2-260-260.259-c0-0-1
Degree $2$
Conductor $260$
Sign $1$
Analytic cond. $0.129756$
Root an. cond. $0.360217$
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 + 4-s − 5-s + 8-s − 9-s − 10-s − 13-s + 16-s − 18-s − 20-s + 25-s − 26-s − 2·29-s + 32-s − 36-s + 2·37-s − 40-s + 45-s + 49-s + 50-s − 52-s − 2·58-s − 2·61-s + 64-s + 65-s − 72-s + 2·73-s + ⋯
L(s)  = 1  + 2-s + 4-s − 5-s + 8-s − 9-s − 10-s − 13-s + 16-s − 18-s − 20-s + 25-s − 26-s − 2·29-s + 32-s − 36-s + 2·37-s − 40-s + 45-s + 49-s + 50-s − 52-s − 2·58-s − 2·61-s + 64-s + 65-s − 72-s + 2·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $1$
Analytic conductor: \(0.129756\)
Root analytic conductor: \(0.360217\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{260} (259, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.087923869\)
\(L(\frac12)\) \(\approx\) \(1.087923869\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 + T \)
13 \( 1 + T \)
good3 \( 1 + T^{2} \)
7 \( ( 1 - T )( 1 + T ) \)
11 \( 1 + T^{2} \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( 1 + T^{2} \)
23 \( 1 + T^{2} \)
29 \( ( 1 + T )^{2} \)
31 \( 1 + T^{2} \)
37 \( ( 1 - T )^{2} \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( 1 + T^{2} \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( 1 + T^{2} \)
61 \( ( 1 + T )^{2} \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( 1 + T^{2} \)
73 \( ( 1 - T )^{2} \)
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

−12.19967017774479780057906559557, −11.46147995824354979587019963387, −10.80587212636890756918393946839, −9.360420499424939234268191907151, −7.986341427730437197763434823045, −7.29729982719558880608976331354, −6.00497753991341699599612703706, −4.92792609964486891482161755115, −3.79583774710055103614775868217, −2.60456607303142835073975488166, 2.60456607303142835073975488166, 3.79583774710055103614775868217, 4.92792609964486891482161755115, 6.00497753991341699599612703706, 7.29729982719558880608976331354, 7.986341427730437197763434823045, 9.360420499424939234268191907151, 10.80587212636890756918393946839, 11.46147995824354979587019963387, 12.19967017774479780057906559557

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