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Properties

Label 2-1300-52.35-c0-0-0
Degree $2$
Conductor $1300$
Sign $-0.711 + 0.702i$
Analytic cond. $0.648784$
Root an. cond. $0.805471$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s − 0.999·8-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)13-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s − 0.999·18-s + (−0.499 − 0.866i)26-s + (0.5 − 0.866i)29-s + (0.499 + 0.866i)32-s − 0.999·34-s + (−0.499 + 0.866i)36-s + (−0.5 + 0.866i)37-s + (0.5 − 0.866i)41-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s − 0.999·8-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)13-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s − 0.999·18-s + (−0.499 − 0.866i)26-s + (0.5 − 0.866i)29-s + (0.499 + 0.866i)32-s − 0.999·34-s + (−0.499 + 0.866i)36-s + (−0.5 + 0.866i)37-s + (0.5 − 0.866i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1300\)    =    \(2^{2} \cdot 5^{2} \cdot 13\)
Sign: $-0.711 + 0.702i$
Analytic conductor: \(0.648784\)
Root analytic conductor: \(0.805471\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1300} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1300,\ (\ :0),\ -0.711 + 0.702i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.150581730\)
\(L(\frac12)\) \(\approx\) \(1.150581730\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 \)
13 \( 1 + (-0.5 + 0.866i)T \)
good3 \( 1 + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T + T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.643459345138870173566842067214, −8.942944284633502120401481000426, −8.164929632129156754462801537614, −6.85488248620044246365419529149, −6.01969890404621571585186463490, −5.26697232213288397708490348288, −4.21909839307984958885684191902, −3.30123926838938206817027822151, −2.46938913889263881808019186782, −0.856459917117394552281502224461, 2.08772791411382502850709799067, 3.37975533014347964290218458648, 4.33823457292943185879583522982, 5.15847957699909673059212022035, 6.04964774932371898863144009287, 6.77549554023365768424715301641, 7.64824748595528663401383321329, 8.530919110704119651830607033475, 8.918542431343625099747325996942, 10.09606082358940032598558818559

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