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Properties

Label 2-13-13.12-c11-0-6
Degree $2$
Conductor $13$
Sign $0.832 - 0.553i$
Analytic cond. $9.98846$
Root an. cond. $3.16045$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.38i·2-s + 494.·3-s + 2.03e3·4-s + 9.09e3i·5-s + 1.67e3i·6-s − 4.50e4i·7-s + 1.38e4i·8-s + 6.74e4·9-s − 3.07e4·10-s + 8.10e5i·11-s + 1.00e6·12-s + (1.11e6 − 7.41e5i)13-s + 1.52e5·14-s + 4.49e6i·15-s + 4.12e6·16-s − 8.51e5·17-s + ⋯
L(s)  = 1  + 0.0748i·2-s + 1.17·3-s + 0.994·4-s + 1.30i·5-s + 0.0879i·6-s − 1.01i·7-s + 0.149i·8-s + 0.380·9-s − 0.0973·10-s + 1.51i·11-s + 1.16·12-s + (0.832 − 0.553i)13-s + 0.0757·14-s + 1.52i·15-s + 0.983·16-s − 0.145·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $0.832 - 0.553i$
Analytic conductor: \(9.98846\)
Root analytic conductor: \(3.16045\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{13} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 13,\ (\ :11/2),\ 0.832 - 0.553i)\)

Particular Values

\(L(6)\) \(\approx\) \(2.87606 + 0.869156i\)
\(L(\frac12)\) \(\approx\) \(2.87606 + 0.869156i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-1.11e6 + 7.41e5i)T \)
good2 \( 1 - 3.38iT - 2.04e3T^{2} \)
3 \( 1 - 494.T + 1.77e5T^{2} \)
5 \( 1 - 9.09e3iT - 4.88e7T^{2} \)
7 \( 1 + 4.50e4iT - 1.97e9T^{2} \)
11 \( 1 - 8.10e5iT - 2.85e11T^{2} \)
17 \( 1 + 8.51e5T + 3.42e13T^{2} \)
19 \( 1 + 7.56e6iT - 1.16e14T^{2} \)
23 \( 1 - 3.69e6T + 9.52e14T^{2} \)
29 \( 1 + 9.33e7T + 1.22e16T^{2} \)
31 \( 1 + 2.61e8iT - 2.54e16T^{2} \)
37 \( 1 - 5.38e8iT - 1.77e17T^{2} \)
41 \( 1 + 6.71e8iT - 5.50e17T^{2} \)
43 \( 1 + 1.09e9T + 9.29e17T^{2} \)
47 \( 1 + 1.79e9iT - 2.47e18T^{2} \)
53 \( 1 + 5.82e9T + 9.26e18T^{2} \)
59 \( 1 + 4.74e9iT - 3.01e19T^{2} \)
61 \( 1 - 6.50e9T + 4.35e19T^{2} \)
67 \( 1 - 8.11e9iT - 1.22e20T^{2} \)
71 \( 1 + 3.16e9iT - 2.31e20T^{2} \)
73 \( 1 + 8.22e9iT - 3.13e20T^{2} \)
79 \( 1 + 3.75e9T + 7.47e20T^{2} \)
83 \( 1 + 6.60e9iT - 1.28e21T^{2} \)
89 \( 1 - 5.68e10iT - 2.77e21T^{2} \)
97 \( 1 - 1.62e11iT - 7.15e21T^{2} \)
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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

−17.29906714210752224394252443066, −15.35420408086869828475349105370, −14.76652927694333747793908477313, −13.37798606166177267107545640837, −11.20269806544157969933082503823, −10.01331046275190010425200154058, −7.74021411301937632564328166971, −6.78600518341283026553729906378, −3.47995676581006189711868989029, −2.15378490081142466090723972204, 1.58372886400560506170975732087, 3.24965339093912644572060283925, 5.84188689196821682929468046942, 8.270291689183220044702935800897, 9.020132897236643151964337723998, 11.36703659709880192901164020079, 12.77866933116500210944351244313, 14.25310855288258935172935309164, 15.77192278623950184810759810177, 16.53989543056150357899887484633

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