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Properties

Label 2-13-13.12-c7-0-3
Degree $2$
Conductor $13$
Sign $0.158 + 0.987i$
Analytic cond. $4.06100$
Root an. cond. $2.01519$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.43i·2-s − 51.4·3-s + 122.·4-s − 488. i·5-s − 125. i·6-s − 616. i·7-s + 609. i·8-s + 460.·9-s + 1.19e3·10-s − 4.62e3i·11-s − 6.27e3·12-s + (1.25e3 + 7.82e3i)13-s + 1.50e3·14-s + 2.51e4i·15-s + 1.41e4·16-s − 1.95e4·17-s + ⋯
L(s)  = 1  + 0.215i·2-s − 1.10·3-s + 0.953·4-s − 1.74i·5-s − 0.237i·6-s − 0.679i·7-s + 0.421i·8-s + 0.210·9-s + 0.376·10-s − 1.04i·11-s − 1.04·12-s + (0.158 + 0.987i)13-s + 0.146·14-s + 1.92i·15-s + 0.862·16-s − 0.964·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.158 + 0.987i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.158 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $0.158 + 0.987i$
Analytic conductor: \(4.06100\)
Root analytic conductor: \(2.01519\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{13} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 13,\ (\ :7/2),\ 0.158 + 0.987i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.856655 - 0.729892i\)
\(L(\frac12)\) \(\approx\) \(0.856655 - 0.729892i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-1.25e3 - 7.82e3i)T \)
good2 \( 1 - 2.43iT - 128T^{2} \)
3 \( 1 + 51.4T + 2.18e3T^{2} \)
5 \( 1 + 488. iT - 7.81e4T^{2} \)
7 \( 1 + 616. iT - 8.23e5T^{2} \)
11 \( 1 + 4.62e3iT - 1.94e7T^{2} \)
17 \( 1 + 1.95e4T + 4.10e8T^{2} \)
19 \( 1 + 2.00e4iT - 8.93e8T^{2} \)
23 \( 1 - 7.13e4T + 3.40e9T^{2} \)
29 \( 1 - 1.38e5T + 1.72e10T^{2} \)
31 \( 1 - 1.82e4iT - 2.75e10T^{2} \)
37 \( 1 - 4.46e5iT - 9.49e10T^{2} \)
41 \( 1 + 1.01e5iT - 1.94e11T^{2} \)
43 \( 1 - 1.03e5T + 2.71e11T^{2} \)
47 \( 1 + 4.45e5iT - 5.06e11T^{2} \)
53 \( 1 - 2.03e5T + 1.17e12T^{2} \)
59 \( 1 + 2.50e6iT - 2.48e12T^{2} \)
61 \( 1 + 7.92e5T + 3.14e12T^{2} \)
67 \( 1 - 3.17e6iT - 6.06e12T^{2} \)
71 \( 1 + 1.85e6iT - 9.09e12T^{2} \)
73 \( 1 + 4.25e6iT - 1.10e13T^{2} \)
79 \( 1 - 2.38e6T + 1.92e13T^{2} \)
83 \( 1 - 1.12e6iT - 2.71e13T^{2} \)
89 \( 1 - 8.00e6iT - 4.42e13T^{2} \)
97 \( 1 - 2.76e5iT - 8.07e13T^{2} \)
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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

−17.24396770375373292297983492156, −16.69891247717502873263501738154, −15.86620409968377568367558924963, −13.48601263885551883608017438425, −11.95426671980935716545409972903, −10.99601314291094018469338709872, −8.679170094350842031203687605705, −6.52548784695551648180895337821, −4.93666054130153860427867968019, −0.892437544707419309294015539528, 2.65992358500043572121106935725, 5.98959406007401701645779062436, 7.13524546219852640362098146785, 10.39693235349534933812910087056, 11.15844309924507073768099799877, 12.35657894165220888145219443891, 14.81883537170663425290174441233, 15.66713083403048974237596507608, 17.47763646805962315202017547987, 18.37574127409755957438427880844

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