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Properties

Label 2-13-13.12-c13-0-4
Degree $2$
Conductor $13$
Sign $-0.932 - 0.361i$
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  + 110. i·2-s + 787.·3-s − 3.92e3·4-s − 2.87e4i·5-s + 8.67e4i·6-s + 3.70e5i·7-s + 4.69e5i·8-s − 9.73e5·9-s + 3.16e6·10-s + 6.59e6i·11-s − 3.09e6·12-s + (1.62e7 + 6.29e6i)13-s − 4.08e7·14-s − 2.26e7i·15-s − 8.38e7·16-s − 1.44e8·17-s + ⋯
L(s)  = 1  + 1.21i·2-s + 0.623·3-s − 0.479·4-s − 0.823i·5-s + 0.758i·6-s + 1.19i·7-s + 0.633i·8-s − 0.610·9-s + 1.00·10-s + 1.12i·11-s − 0.299·12-s + (0.932 + 0.361i)13-s − 1.44·14-s − 0.514i·15-s − 1.24·16-s − 1.45·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.932 - 0.361i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.932 - 0.361i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $-0.932 - 0.361i$
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: \((2,\ 13,\ (\ :13/2),\ -0.932 - 0.361i)\)

Particular Values

\(L(7)\) \(\approx\) \(0.357167 + 1.90922i\)
\(L(\frac12)\) \(\approx\) \(0.357167 + 1.90922i\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-1.62e7 - 6.29e6i)T \)
good2 \( 1 - 110. iT - 8.19e3T^{2} \)
3 \( 1 - 787.T + 1.59e6T^{2} \)
5 \( 1 + 2.87e4iT - 1.22e9T^{2} \)
7 \( 1 - 3.70e5iT - 9.68e10T^{2} \)
11 \( 1 - 6.59e6iT - 3.45e13T^{2} \)
17 \( 1 + 1.44e8T + 9.90e15T^{2} \)
19 \( 1 - 6.36e7iT - 4.20e16T^{2} \)
23 \( 1 - 3.74e8T + 5.04e17T^{2} \)
29 \( 1 - 4.23e9T + 1.02e19T^{2} \)
31 \( 1 + 4.92e9iT - 2.44e19T^{2} \)
37 \( 1 - 2.27e10iT - 2.43e20T^{2} \)
41 \( 1 - 1.61e10iT - 9.25e20T^{2} \)
43 \( 1 + 1.62e10T + 1.71e21T^{2} \)
47 \( 1 + 7.34e10iT - 5.46e21T^{2} \)
53 \( 1 - 6.38e10T + 2.60e22T^{2} \)
59 \( 1 - 1.09e11iT - 1.04e23T^{2} \)
61 \( 1 - 6.41e11T + 1.61e23T^{2} \)
67 \( 1 + 4.51e11iT - 5.48e23T^{2} \)
71 \( 1 + 1.41e12iT - 1.16e24T^{2} \)
73 \( 1 - 1.80e11iT - 1.67e24T^{2} \)
79 \( 1 - 3.28e12T + 4.66e24T^{2} \)
83 \( 1 - 1.66e12iT - 8.87e24T^{2} \)
89 \( 1 + 6.68e12iT - 2.19e25T^{2} \)
97 \( 1 - 1.69e12iT - 6.73e25T^{2} \)
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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

−16.95974391455892872262211836832, −15.63277524201187613959394204283, −14.90252268098627390714056897100, −13.38268536663445215435970740409, −11.71569223756969737306724146811, −9.010626353909122025615573851400, −8.340869693992957136886967531642, −6.42136718947232124281588926013, −4.90178286988232730803242381368, −2.24008329997840964039107368005, 0.73926801567107482093439016659, 2.70194640772772156882016514444, 3.71651114882558683697861390457, 6.74405151359136037238050497382, 8.737533317430541574456033464493, 10.65900090435168696198462010612, 11.15113692691908888885907897059, 13.26892839768944761200924926979, 14.13744032440233731922770259306, 15.92119355406019130790476925907

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