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Properties

Label 2-13-13.12-c13-0-5
Degree $2$
Conductor $13$
Sign $0.564 - 0.825i$
Analytic cond. $13.9400$
Root an. cond. $3.73363$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 28.8i·2-s + 701.·3-s + 7.36e3·4-s + 3.58e4i·5-s − 2.02e4i·6-s + 3.18e5i·7-s − 4.48e5i·8-s − 1.10e6·9-s + 1.03e6·10-s + 1.85e6i·11-s + 5.16e6·12-s + (−9.82e6 + 1.43e7i)13-s + 9.18e6·14-s + 2.51e7i·15-s + 4.73e7·16-s + 1.16e8·17-s + ⋯
L(s)  = 1  − 0.318i·2-s + 0.555·3-s + 0.898·4-s + 1.02i·5-s − 0.177i·6-s + 1.02i·7-s − 0.605i·8-s − 0.691·9-s + 0.327·10-s + 0.314i·11-s + 0.499·12-s + (−0.564 + 0.825i)13-s + 0.325·14-s + 0.570i·15-s + 0.705·16-s + 1.16·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.564 - 0.825i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.564 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $0.564 - 0.825i$
Analytic conductor: \(13.9400\)
Root analytic conductor: \(3.73363\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{13} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 13,\ (\ :13/2),\ 0.564 - 0.825i)\)

Particular Values

\(L(7)\) \(\approx\) \(2.16301 + 1.14137i\)
\(L(\frac12)\) \(\approx\) \(2.16301 + 1.14137i\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (9.82e6 - 1.43e7i)T \)
good2 \( 1 + 28.8iT - 8.19e3T^{2} \)
3 \( 1 - 701.T + 1.59e6T^{2} \)
5 \( 1 - 3.58e4iT - 1.22e9T^{2} \)
7 \( 1 - 3.18e5iT - 9.68e10T^{2} \)
11 \( 1 - 1.85e6iT - 3.45e13T^{2} \)
17 \( 1 - 1.16e8T + 9.90e15T^{2} \)
19 \( 1 - 1.45e8iT - 4.20e16T^{2} \)
23 \( 1 - 5.16e8T + 5.04e17T^{2} \)
29 \( 1 - 2.46e8T + 1.02e19T^{2} \)
31 \( 1 + 7.87e9iT - 2.44e19T^{2} \)
37 \( 1 + 1.76e10iT - 2.43e20T^{2} \)
41 \( 1 - 4.12e10iT - 9.25e20T^{2} \)
43 \( 1 - 1.04e10T + 1.71e21T^{2} \)
47 \( 1 - 7.32e10iT - 5.46e21T^{2} \)
53 \( 1 - 1.17e11T + 2.60e22T^{2} \)
59 \( 1 + 6.02e11iT - 1.04e23T^{2} \)
61 \( 1 + 7.81e11T + 1.61e23T^{2} \)
67 \( 1 + 4.22e11iT - 5.48e23T^{2} \)
71 \( 1 - 3.81e11iT - 1.16e24T^{2} \)
73 \( 1 + 6.03e11iT - 1.67e24T^{2} \)
79 \( 1 - 1.45e12T + 4.66e24T^{2} \)
83 \( 1 + 2.97e12iT - 8.87e24T^{2} \)
89 \( 1 - 4.31e12iT - 2.19e25T^{2} \)
97 \( 1 + 8.11e12iT - 6.73e25T^{2} \)
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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

−16.72215381699519693417773756826, −15.09860226219601696050443114725, −14.44490864751071515058084412205, −12.26310851658914129301994666576, −11.16989454335753677669770830976, −9.562461076279187958223626721721, −7.62992280772895639902472563831, −6.05407532966000798811015257885, −3.13048678480197644830064784463, −2.15416940347697775521496472070, 0.966279636849123819458032464889, 3.05414178101646736662403005057, 5.32137734019592218454852056076, 7.33153380226744297898061739083, 8.602973446900382889673447944065, 10.53263883731639284284871554604, 12.18624217520174493548857170548, 13.75231385372235399495366483299, 15.06988099468208445700029357466, 16.54586720849060700023844053351

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