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Properties

Label 2-1300-65.33-c1-0-7
Degree $2$
Conductor $1300$
Sign $0.999 + 0.00863i$
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  + (−1.86 + 0.5i)3-s + (−3.23 + 1.86i)7-s + (0.633 − 0.366i)9-s + (−0.598 − 2.23i)11-s + (−3 − 2i)13-s + (1.13 − 4.23i)17-s + (0.866 + 0.232i)19-s + (5.09 − 5.09i)21-s + (1.86 + 6.96i)23-s + (3.09 − 3.09i)27-s + (−7.96 − 4.59i)29-s + (5.73 + 5.73i)31-s + (2.23 + 3.86i)33-s + (−0.232 − 0.133i)37-s + (6.59 + 2.23i)39-s + ⋯
L(s)  = 1  + (−1.07 + 0.288i)3-s + (−1.22 + 0.705i)7-s + (0.211 − 0.122i)9-s + (−0.180 − 0.672i)11-s + (−0.832 − 0.554i)13-s + (0.275 − 1.02i)17-s + (0.198 + 0.0532i)19-s + (1.11 − 1.11i)21-s + (0.389 + 1.45i)23-s + (0.596 − 0.596i)27-s + (−1.47 − 0.853i)29-s + (1.02 + 1.02i)31-s + (0.388 + 0.672i)33-s + (−0.0381 − 0.0220i)37-s + (1.05 + 0.357i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00863i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1300\)    =    \(2^{2} \cdot 5^{2} \cdot 13\)
Sign: $0.999 + 0.00863i$
Analytic conductor: \(10.3805\)
Root analytic conductor: \(3.22188\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1300} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1300,\ (\ :1/2),\ 0.999 + 0.00863i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6614747232\)
\(L(\frac12)\) \(\approx\) \(0.6614747232\)
\(L(\frac{3}{2})\) 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 \( 1 + (3 + 2i)T \)
good3 \( 1 + (1.86 - 0.5i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (3.23 - 1.86i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.598 + 2.23i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.13 + 4.23i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.866 - 0.232i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.86 - 6.96i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (7.96 + 4.59i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.73 - 5.73i)T + 31iT^{2} \)
37 \( 1 + (0.232 + 0.133i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.133 - 0.0358i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-11.3 - 3.03i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + 0.535iT - 47T^{2} \)
53 \( 1 + (-1.53 - 1.53i)T + 53iT^{2} \)
59 \( 1 + (1.79 - 6.69i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.76 + 4.79i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.13 + 4.23i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 - 12.9T + 73T^{2} \)
79 \( 1 + 4.53iT - 79T^{2} \)
83 \( 1 + 3.46iT - 83T^{2} \)
89 \( 1 + (-14.7 + 3.96i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-2.23 - 3.86i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.608588011090597613861728155150, −9.194360866212169430706060171189, −7.917547503424052962094879936634, −7.08689918814279099649722227671, −6.01382517793420824736925898841, −5.61518955941947737368383292993, −4.83900088794655565949423611644, −3.40231329252299899968277041360, −2.62870579177595165628263388708, −0.54049740520102539802482890666, 0.67344382768063326998016782029, 2.33068990079707755171376170022, 3.65950547999373006432359031285, 4.59925732960216870554615254772, 5.59898337339426222436416122728, 6.47588285234550885081584999640, 6.92531356498586834594752233137, 7.77420815570351676002035867030, 9.040859093339017248187718937235, 9.799517416463216017641566522023

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