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Properties

Label 2-1264-79.78-c0-0-1
Degree $2$
Conductor $1264$
Sign $1$
Analytic cond. $0.630818$
Root an. cond. $0.794240$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.618·5-s + 9-s + 1.61·11-s − 1.61·13-s − 0.618·19-s − 0.618·23-s − 0.618·25-s + 1.61·31-s + 0.618·45-s + 49-s + 1.00·55-s − 1.00·65-s − 0.618·67-s + 0.618·73-s − 79-s + 81-s − 2·83-s − 1.61·89-s − 0.381·95-s + 0.618·97-s + 1.61·99-s − 1.61·101-s − 0.381·115-s − 1.61·117-s + ⋯
L(s)  = 1  + 0.618·5-s + 9-s + 1.61·11-s − 1.61·13-s − 0.618·19-s − 0.618·23-s − 0.618·25-s + 1.61·31-s + 0.618·45-s + 49-s + 1.00·55-s − 1.00·65-s − 0.618·67-s + 0.618·73-s − 79-s + 81-s − 2·83-s − 1.61·89-s − 0.381·95-s + 0.618·97-s + 1.61·99-s − 1.61·101-s − 0.381·115-s − 1.61·117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1264\)    =    \(2^{4} \cdot 79\)
Sign: $1$
Analytic conductor: \(0.630818\)
Root analytic conductor: \(0.794240\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1264} (1105, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1264,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.270218370\)
\(L(\frac12)\) \(\approx\) \(1.270218370\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
79 \( 1 + T \)
good3 \( 1 - T^{2} \)
5 \( 1 - 0.618T + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - 1.61T + T^{2} \)
13 \( 1 + 1.61T + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + 0.618T + T^{2} \)
23 \( 1 + 0.618T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - 1.61T + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + 0.618T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 0.618T + T^{2} \)
83 \( 1 + 2T + T^{2} \)
89 \( 1 + 1.61T + T^{2} \)
97 \( 1 - 0.618T + T^{2} \)
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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

−9.829801807003953157478102487683, −9.325465599260351693255651923396, −8.294433615856057765424289976892, −7.25544407963965417200099046680, −6.65483324343715194501014118334, −5.80660239597333349970510981606, −4.60149991100736925799271846309, −4.01921444231014817335883723546, −2.51719698111865183022931018397, −1.48750417157765528420555652390, 1.48750417157765528420555652390, 2.51719698111865183022931018397, 4.01921444231014817335883723546, 4.60149991100736925799271846309, 5.80660239597333349970510981606, 6.65483324343715194501014118334, 7.25544407963965417200099046680, 8.294433615856057765424289976892, 9.325465599260351693255651923396, 9.829801807003953157478102487683

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