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Properties

Label 2-1300-65.32-c1-0-4
Degree $2$
Conductor $1300$
Sign $0.993 - 0.110i$
Analytic cond. $10.3805$
Root an. cond. $3.22188$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.877 − 3.27i)3-s + (−2.27 + 1.31i)7-s + (−7.34 + 4.24i)9-s + (1.71 − 0.458i)11-s + (−1.21 + 3.39i)13-s + (−0.363 − 0.0972i)17-s + (−0.462 + 1.72i)19-s + (6.28 + 6.28i)21-s + (4.72 − 1.26i)23-s + (13.1 + 13.1i)27-s + (6.28 + 3.62i)29-s + (2.98 − 2.98i)31-s + (−3.00 − 5.20i)33-s + (−8.83 − 5.10i)37-s + (12.1 + 1.00i)39-s + ⋯
L(s)  = 1  + (−0.506 − 1.88i)3-s + (−0.859 + 0.495i)7-s + (−2.44 + 1.41i)9-s + (0.516 − 0.138i)11-s + (−0.337 + 0.941i)13-s + (−0.0880 − 0.0235i)17-s + (−0.106 + 0.395i)19-s + (1.37 + 1.37i)21-s + (0.986 − 0.264i)23-s + (2.52 + 2.52i)27-s + (1.16 + 0.674i)29-s + (0.535 − 0.535i)31-s + (−0.522 − 0.905i)33-s + (−1.45 − 0.838i)37-s + (1.94 + 0.161i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.110i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1300\)    =    \(2^{2} \cdot 5^{2} \cdot 13\)
Sign: $0.993 - 0.110i$
Analytic conductor: \(10.3805\)
Root analytic conductor: \(3.22188\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1300} (357, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1300,\ (\ :1/2),\ 0.993 - 0.110i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8066417258\)
\(L(\frac12)\) \(\approx\) \(0.8066417258\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 + (1.21 - 3.39i)T \)
good3 \( 1 + (0.877 + 3.27i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (2.27 - 1.31i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.71 + 0.458i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (0.363 + 0.0972i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.462 - 1.72i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-4.72 + 1.26i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-6.28 - 3.62i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.98 + 2.98i)T - 31iT^{2} \)
37 \( 1 + (8.83 + 5.10i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.59 - 5.96i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (0.427 - 1.59i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 - 1.19iT - 47T^{2} \)
53 \( 1 + (5.13 - 5.13i)T - 53iT^{2} \)
59 \( 1 + (2.89 + 0.775i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (2.45 + 4.25i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.98 - 6.90i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.02 - 1.07i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 - 11.2T + 73T^{2} \)
79 \( 1 + 2.72iT - 79T^{2} \)
83 \( 1 - 11.2iT - 83T^{2} \)
89 \( 1 + (-1.91 - 7.14i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-0.838 - 1.45i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.432596350185430671015812626811, −8.737992337923695331478749553107, −7.907331876745327217602686219298, −6.90541413540966467436799794011, −6.57398246582896089171495362645, −5.85977935211013034052008050277, −4.80770704928182806247789601929, −3.15462883477834424727857624290, −2.22574923921121322987017662529, −1.10260208007439029240939361775, 0.41571491638685170732068968339, 2.99052854156876072913682677780, 3.54665241985357652281357001071, 4.58087774786319690233152125374, 5.18235017690949235296303031264, 6.18740847270642559856530336527, 6.93356445239556527029340929156, 8.355006350817971422850031158719, 9.097287912575815140262359566449, 9.823597127405531868422103331688

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