Properties

Label 2-260-1.1-c1-0-3
Degree $2$
Conductor $260$
Sign $1$
Analytic cond. $2.07611$
Root an. cond. $1.44087$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.32·3-s + 5-s − 5.02·7-s + 8.02·9-s + 1.70·11-s + 13-s + 3.32·15-s − 4.64·17-s − 4.34·19-s − 16.6·21-s − 0.679·23-s + 25-s + 16.6·27-s − 1.02·29-s − 2.29·31-s + 5.67·33-s − 5.02·35-s − 1.61·37-s + 3.32·39-s + 4.64·41-s − 3.32·43-s + 8.02·45-s + 1.02·47-s + 18.2·49-s − 15.4·51-s − 9.41·53-s + 1.70·55-s + ⋯
L(s)  = 1  + 1.91·3-s + 0.447·5-s − 1.90·7-s + 2.67·9-s + 0.514·11-s + 0.277·13-s + 0.857·15-s − 1.12·17-s − 0.997·19-s − 3.64·21-s − 0.141·23-s + 0.200·25-s + 3.21·27-s − 0.190·29-s − 0.411·31-s + 0.987·33-s − 0.849·35-s − 0.265·37-s + 0.531·39-s + 0.724·41-s − 0.506·43-s + 1.19·45-s + 0.149·47-s + 2.61·49-s − 2.15·51-s − 1.29·53-s + 0.230·55-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $1$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.064071583\)
\(L(\frac12)\) \(\approx\) \(2.064071583\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T \)
13 \( 1 - T \)
good3 \( 1 - 3.32T + 3T^{2} \)
7 \( 1 + 5.02T + 7T^{2} \)
11 \( 1 - 1.70T + 11T^{2} \)
17 \( 1 + 4.64T + 17T^{2} \)
19 \( 1 + 4.34T + 19T^{2} \)
23 \( 1 + 0.679T + 23T^{2} \)
29 \( 1 + 1.02T + 29T^{2} \)
31 \( 1 + 2.29T + 31T^{2} \)
37 \( 1 + 1.61T + 37T^{2} \)
41 \( 1 - 4.64T + 41T^{2} \)
43 \( 1 + 3.32T + 43T^{2} \)
47 \( 1 - 1.02T + 47T^{2} \)
53 \( 1 + 9.41T + 53T^{2} \)
59 \( 1 + 8.93T + 59T^{2} \)
61 \( 1 - 9.02T + 61T^{2} \)
67 \( 1 - 5.61T + 67T^{2} \)
71 \( 1 - 1.70T + 71T^{2} \)
73 \( 1 - 12.4T + 73T^{2} \)
79 \( 1 + 2.64T + 79T^{2} \)
83 \( 1 + 8.25T + 83T^{2} \)
89 \( 1 - 1.22T + 89T^{2} \)
97 \( 1 + 0.0565T + 97T^{2} \)
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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

−12.57566187362363995259752978564, −10.66052080806835428498984529416, −9.635120035701252929684526255163, −9.223965415667874847638279434336, −8.386265277994660403410791482109, −7.01140185229377012753031027012, −6.34777905280032026341403151687, −4.15995559979417781398737110037, −3.25659101273538783395765151540, −2.17160924833122862384495920329, 2.17160924833122862384495920329, 3.25659101273538783395765151540, 4.15995559979417781398737110037, 6.34777905280032026341403151687, 7.01140185229377012753031027012, 8.386265277994660403410791482109, 9.223965415667874847638279434336, 9.635120035701252929684526255163, 10.66052080806835428498984529416, 12.57566187362363995259752978564

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