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Properties

Label 2-13-13.12-c11-0-7
Degree $2$
Conductor $13$
Sign $0.855 + 0.518i$
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  + 65.0i·2-s + 15.7·3-s − 2.18e3·4-s − 6.67e3i·5-s + 1.02e3i·6-s − 8.15e4i·7-s − 9.17e3i·8-s − 1.76e5·9-s + 4.34e5·10-s − 3.04e5i·11-s − 3.44e4·12-s + (1.14e6 + 6.93e5i)13-s + 5.31e6·14-s − 1.05e5i·15-s − 3.88e6·16-s + 6.40e5·17-s + ⋯
L(s)  = 1  + 1.43i·2-s + 0.0374·3-s − 1.06·4-s − 0.955i·5-s + 0.0537i·6-s − 1.83i·7-s − 0.0990i·8-s − 0.998·9-s + 1.37·10-s − 0.569i·11-s − 0.0399·12-s + (0.855 + 0.518i)13-s + 2.63·14-s − 0.0357i·15-s − 0.926·16-s + 0.109·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.855 + 0.518i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.855 + 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $0.855 + 0.518i$
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: \((2,\ 13,\ (\ :11/2),\ 0.855 + 0.518i)\)

Particular Values

\(L(6)\) \(\approx\) \(1.14174 - 0.318956i\)
\(L(\frac12)\) \(\approx\) \(1.14174 - 0.318956i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-1.14e6 - 6.93e5i)T \)
good2 \( 1 - 65.0iT - 2.04e3T^{2} \)
3 \( 1 - 15.7T + 1.77e5T^{2} \)
5 \( 1 + 6.67e3iT - 4.88e7T^{2} \)
7 \( 1 + 8.15e4iT - 1.97e9T^{2} \)
11 \( 1 + 3.04e5iT - 2.85e11T^{2} \)
17 \( 1 - 6.40e5T + 3.42e13T^{2} \)
19 \( 1 + 1.18e7iT - 1.16e14T^{2} \)
23 \( 1 + 4.56e7T + 9.52e14T^{2} \)
29 \( 1 - 4.74e6T + 1.22e16T^{2} \)
31 \( 1 + 1.52e8iT - 2.54e16T^{2} \)
37 \( 1 + 2.70e8iT - 1.77e17T^{2} \)
41 \( 1 - 9.31e7iT - 5.50e17T^{2} \)
43 \( 1 - 1.44e9T + 9.29e17T^{2} \)
47 \( 1 - 2.06e9iT - 2.47e18T^{2} \)
53 \( 1 - 2.76e9T + 9.26e18T^{2} \)
59 \( 1 - 2.69e9iT - 3.01e19T^{2} \)
61 \( 1 + 6.51e9T + 4.35e19T^{2} \)
67 \( 1 - 9.67e8iT - 1.22e20T^{2} \)
71 \( 1 + 1.34e10iT - 2.31e20T^{2} \)
73 \( 1 - 8.89e8iT - 3.13e20T^{2} \)
79 \( 1 + 3.87e10T + 7.47e20T^{2} \)
83 \( 1 + 4.24e10iT - 1.28e21T^{2} \)
89 \( 1 - 8.88e10iT - 2.77e21T^{2} \)
97 \( 1 - 3.90e10iT - 7.15e21T^{2} \)
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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

−16.75103783407817042145974298859, −16.07944938027978650511739802507, −14.18681057234184785903929380358, −13.48938680076222710859403899592, −11.13363852145482943940462022222, −8.903482270526351574645275830249, −7.66876477778276440579983784220, −6.06250566259056661492474082009, −4.32096751900687701293432920540, −0.54927438042987742222612485681, 2.12465564517537940798321438799, 3.22960541421878016950367116764, 5.92209049908925106449836321969, 8.641677932962148471216790689843, 10.22380193224514393497120575658, 11.56020547049403327260929867617, 12.43425993427400235841435623057, 14.31241927131628802386153934918, 15.63693841919213883361629185942, 18.09933438293818336751260715443

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