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Properties

Label 2-1300-13.10-c1-0-7
Degree $2$
Conductor $1300$
Sign $0.964 - 0.265i$
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  + (−0.5 − 0.866i)3-s + (1.5 + 0.866i)7-s + (1 − 1.73i)9-s + (−4.5 + 2.59i)11-s + (1 + 3.46i)13-s + (1.5 − 2.59i)17-s + (4.5 + 2.59i)19-s − 1.73i·21-s + (1.5 + 2.59i)23-s − 5·27-s + (4.5 + 7.79i)29-s − 3.46i·31-s + (4.5 + 2.59i)33-s + (4.5 − 2.59i)37-s + (2.49 − 2.59i)39-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (0.566 + 0.327i)7-s + (0.333 − 0.577i)9-s + (−1.35 + 0.783i)11-s + (0.277 + 0.960i)13-s + (0.363 − 0.630i)17-s + (1.03 + 0.596i)19-s − 0.377i·21-s + (0.312 + 0.541i)23-s − 0.962·27-s + (0.835 + 1.44i)29-s − 0.622i·31-s + (0.783 + 0.452i)33-s + (0.739 − 0.427i)37-s + (0.400 − 0.416i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1300\)    =    \(2^{2} \cdot 5^{2} \cdot 13\)
Sign: $0.964 - 0.265i$
Analytic conductor: \(10.3805\)
Root analytic conductor: \(3.22188\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1300} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1300,\ (\ :1/2),\ 0.964 - 0.265i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.535047295\)
\(L(\frac12)\) \(\approx\) \(1.535047295\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - 3.46i)T \)
good3 \( 1 + (0.5 + 0.866i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (-1.5 - 0.866i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (4.5 - 2.59i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.5 - 2.59i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.5 - 7.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 + (-4.5 + 2.59i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.5 - 2.59i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.5 + 4.33i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 10.3iT - 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (-4.5 - 2.59i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.5 + 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.5 - 0.866i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.5 - 2.59i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 6.92iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 10.3iT - 83T^{2} \)
89 \( 1 + (-13.5 + 7.79i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-10.5 - 6.06i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.660949538743259093517341498493, −8.949595634199294107601618680668, −7.79000730945903400372992087616, −7.36258764965649265486824886480, −6.45758364719040106128559207511, −5.41780966526619998935576111644, −4.78938274234972267405263228966, −3.53470829124263434352025341716, −2.28649117451603650638651569807, −1.16868972636593581319277654011, 0.799829864379779536076872081870, 2.46956968065341283427625588496, 3.49953181558041287575315949395, 4.71610718367543041920046744696, 5.26187519485461421089403378375, 6.07688079731003215530567131724, 7.40748570084098382309495584313, 8.006249604143091966761410720740, 8.613279936694260602155544632297, 10.03696073390117090096506008163

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