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Properties

Label 2-1300-65.2-c1-0-12
Degree $2$
Conductor $1300$
Sign $-0.201 + 0.979i$
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  + (−2.49 − 0.668i)3-s + (0.749 + 0.432i)7-s + (3.17 + 1.83i)9-s + (0.417 − 1.55i)11-s + (−1.32 − 3.35i)13-s + (1.33 + 4.99i)17-s + (−1.16 + 0.312i)19-s + (−1.57 − 1.57i)21-s + (−0.704 + 2.62i)23-s + (−1.20 − 1.20i)27-s + (5.94 − 3.43i)29-s + (0.191 − 0.191i)31-s + (−2.08 + 3.60i)33-s + (8.22 − 4.74i)37-s + (1.06 + 9.24i)39-s + ⋯
L(s)  = 1  + (−1.43 − 0.385i)3-s + (0.283 + 0.163i)7-s + (1.05 + 0.610i)9-s + (0.125 − 0.469i)11-s + (−0.367 − 0.930i)13-s + (0.324 + 1.21i)17-s + (−0.267 + 0.0716i)19-s + (−0.344 − 0.344i)21-s + (−0.146 + 0.548i)23-s + (−0.232 − 0.232i)27-s + (1.10 − 0.636i)29-s + (0.0344 − 0.0344i)31-s + (−0.362 + 0.627i)33-s + (1.35 − 0.780i)37-s + (0.170 + 1.48i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.201 + 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.201 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1300\)    =    \(2^{2} \cdot 5^{2} \cdot 13\)
Sign: $-0.201 + 0.979i$
Analytic conductor: \(10.3805\)
Root analytic conductor: \(3.22188\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1300} (457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1300,\ (\ :1/2),\ -0.201 + 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7267788082\)
\(L(\frac12)\) \(\approx\) \(0.7267788082\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.32 + 3.35i)T \)
good3 \( 1 + (2.49 + 0.668i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (-0.749 - 0.432i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.417 + 1.55i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-1.33 - 4.99i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.16 - 0.312i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.704 - 2.62i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-5.94 + 3.43i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.191 + 0.191i)T - 31iT^{2} \)
37 \( 1 + (-8.22 + 4.74i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.417 - 0.111i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (11.5 - 3.09i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + 1.52iT - 47T^{2} \)
53 \( 1 + (-4.88 + 4.88i)T - 53iT^{2} \)
59 \( 1 + (3.05 + 11.3i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (2.94 - 5.09i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.76 + 11.7i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.85 + 14.3i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 - 10.9T + 73T^{2} \)
79 \( 1 - 3.98iT - 79T^{2} \)
83 \( 1 + 6.25iT - 83T^{2} \)
89 \( 1 + (9.80 + 2.62i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-0.728 + 1.26i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.667590200474468874317923323684, −8.348266435021264114448444148764, −7.84181262068175836194998118517, −6.68371497075148772167057766050, −6.04397830208943543125905044276, −5.41132126175026965622917275079, −4.53350072505375135395591069657, −3.26827092326193069847171379691, −1.72714482462767025645128508190, −0.43328996746068474684822022363, 1.13415582758398046439177721716, 2.71065111015988004208606252155, 4.34867205857567421724132369918, 4.71957062287321389066919984870, 5.62036245350994888942744851657, 6.61604234732048291725890737424, 7.08972375057441635361452853140, 8.254626998661705928565097330248, 9.326349036541815218108840928540, 10.01934647267139532663024363019

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