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Properties

Label 2-13-13.8-c12-0-12
Degree $2$
Conductor $13$
Sign $-0.933 + 0.358i$
Analytic cond. $11.8819$
Root an. cond. $3.44701$
Motivic weight $12$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (21.2 + 21.2i)2-s + 104.·3-s − 3.19e3i·4-s + (−2.68e3 − 2.68e3i)5-s + (2.22e3 + 2.22e3i)6-s + (−1.29e5 + 1.29e5i)7-s + (1.54e5 − 1.54e5i)8-s − 5.20e5·9-s − 1.13e5i·10-s + (−3.44e5 + 3.44e5i)11-s − 3.34e5i·12-s + (−2.96e6 − 3.81e6i)13-s − 5.48e6·14-s + (−2.81e5 − 2.81e5i)15-s − 6.53e6·16-s + 1.83e6i·17-s + ⋯
L(s)  = 1  + (0.331 + 0.331i)2-s + 0.143·3-s − 0.780i·4-s + (−0.171 − 0.171i)5-s + (0.0475 + 0.0475i)6-s + (−1.09 + 1.09i)7-s + (0.589 − 0.589i)8-s − 0.979·9-s − 0.113i·10-s + (−0.194 + 0.194i)11-s − 0.112i·12-s + (−0.613 − 0.789i)13-s − 0.727·14-s + (−0.0247 − 0.0247i)15-s − 0.389·16-s + 0.0759i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 + 0.358i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.933 + 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $-0.933 + 0.358i$
Analytic conductor: \(11.8819\)
Root analytic conductor: \(3.44701\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{13} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 13,\ (\ :6),\ -0.933 + 0.358i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.0604122 - 0.325757i\)
\(L(\frac12)\) \(\approx\) \(0.0604122 - 0.325757i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (2.96e6 + 3.81e6i)T \)
good2 \( 1 + (-21.2 - 21.2i)T + 4.09e3iT^{2} \)
3 \( 1 - 104.T + 5.31e5T^{2} \)
5 \( 1 + (2.68e3 + 2.68e3i)T + 2.44e8iT^{2} \)
7 \( 1 + (1.29e5 - 1.29e5i)T - 1.38e10iT^{2} \)
11 \( 1 + (3.44e5 - 3.44e5i)T - 3.13e12iT^{2} \)
17 \( 1 - 1.83e6iT - 5.82e14T^{2} \)
19 \( 1 + (5.46e7 + 5.46e7i)T + 2.21e15iT^{2} \)
23 \( 1 - 2.17e8iT - 2.19e16T^{2} \)
29 \( 1 - 1.03e9T + 3.53e17T^{2} \)
31 \( 1 + (-7.03e8 - 7.03e8i)T + 7.87e17iT^{2} \)
37 \( 1 + (1.28e9 - 1.28e9i)T - 6.58e18iT^{2} \)
41 \( 1 + (5.38e9 + 5.38e9i)T + 2.25e19iT^{2} \)
43 \( 1 - 3.79e8iT - 3.99e19T^{2} \)
47 \( 1 + (3.34e9 - 3.34e9i)T - 1.16e20iT^{2} \)
53 \( 1 - 2.86e10T + 4.91e20T^{2} \)
59 \( 1 + (-1.04e10 + 1.04e10i)T - 1.77e21iT^{2} \)
61 \( 1 + 5.84e8T + 2.65e21T^{2} \)
67 \( 1 + (1.22e10 + 1.22e10i)T + 8.18e21iT^{2} \)
71 \( 1 + (1.51e11 + 1.51e11i)T + 1.64e22iT^{2} \)
73 \( 1 + (-4.01e10 + 4.01e10i)T - 2.29e22iT^{2} \)
79 \( 1 + 2.06e11T + 5.90e22T^{2} \)
83 \( 1 + (1.61e9 + 1.61e9i)T + 1.06e23iT^{2} \)
89 \( 1 + (2.30e11 - 2.30e11i)T - 2.46e23iT^{2} \)
97 \( 1 + (2.81e8 + 2.81e8i)T + 6.93e23iT^{2} \)
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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

−15.79595147024054110062869045253, −15.11369762935377374004353759482, −13.57367238460978881602332435295, −12.14258768167121274005865130045, −10.21010717492372956710810269293, −8.745044471456518816057909085025, −6.47417815751594627987213084482, −5.18052255757163421657021359446, −2.70931583507960255997829959701, −0.11915546214596388769245304167, 2.77557946457949312815637780197, 4.13380370964427945382985645150, 6.73137100556417446979987396233, 8.375979738023991462198116465708, 10.35933685884377478999987181829, 11.91935183146464790446264937940, 13.22352904065061773077775017336, 14.37299554564057823470569984239, 16.48443019415014909630787310991, 17.08058299748513635035716142329

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