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Properties

Label 2-13-13.5-c4-0-0
Degree $2$
Conductor $13$
Sign $-0.911 - 0.410i$
Analytic cond. $1.34380$
Root an. cond. $1.15922$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.18 + 3.18i)2-s − 10.6·3-s − 4.25i·4-s + (−9.43 + 9.43i)5-s + (33.7 − 33.7i)6-s + (54.8 + 54.8i)7-s + (−37.3 − 37.3i)8-s + 31.6·9-s − 60.0i·10-s + (−12.2 − 12.2i)11-s + 45.1i·12-s + (−111. + 127. i)13-s − 349.·14-s + (100. − 100. i)15-s + 305.·16-s + 327. i·17-s + ⋯
L(s)  = 1  + (−0.795 + 0.795i)2-s − 1.17·3-s − 0.265i·4-s + (−0.377 + 0.377i)5-s + (0.938 − 0.938i)6-s + (1.12 + 1.12i)7-s + (−0.584 − 0.584i)8-s + 0.390·9-s − 0.600i·10-s + (−0.101 − 0.101i)11-s + 0.313i·12-s + (−0.657 + 0.753i)13-s − 1.78·14-s + (0.444 − 0.444i)15-s + 1.19·16-s + 1.13i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.911 - 0.410i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.911 - 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $-0.911 - 0.410i$
Analytic conductor: \(1.34380\)
Root analytic conductor: \(1.15922\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{13} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 13,\ (\ :2),\ -0.911 - 0.410i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0911143 + 0.424005i\)
\(L(\frac12)\) \(\approx\) \(0.0911143 + 0.424005i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (111. - 127. i)T \)
good2 \( 1 + (3.18 - 3.18i)T - 16iT^{2} \)
3 \( 1 + 10.6T + 81T^{2} \)
5 \( 1 + (9.43 - 9.43i)T - 625iT^{2} \)
7 \( 1 + (-54.8 - 54.8i)T + 2.40e3iT^{2} \)
11 \( 1 + (12.2 + 12.2i)T + 1.46e4iT^{2} \)
17 \( 1 - 327. iT - 8.35e4T^{2} \)
19 \( 1 + (-438. + 438. i)T - 1.30e5iT^{2} \)
23 \( 1 - 229. iT - 2.79e5T^{2} \)
29 \( 1 + 443.T + 7.07e5T^{2} \)
31 \( 1 + (424. - 424. i)T - 9.23e5iT^{2} \)
37 \( 1 + (766. + 766. i)T + 1.87e6iT^{2} \)
41 \( 1 + (-1.51e3 + 1.51e3i)T - 2.82e6iT^{2} \)
43 \( 1 + 192. iT - 3.41e6T^{2} \)
47 \( 1 + (-947. - 947. i)T + 4.87e6iT^{2} \)
53 \( 1 + 664.T + 7.89e6T^{2} \)
59 \( 1 + (-3.92e3 - 3.92e3i)T + 1.21e7iT^{2} \)
61 \( 1 - 379.T + 1.38e7T^{2} \)
67 \( 1 + (185. - 185. i)T - 2.01e7iT^{2} \)
71 \( 1 + (537. - 537. i)T - 2.54e7iT^{2} \)
73 \( 1 + (-992. - 992. i)T + 2.83e7iT^{2} \)
79 \( 1 + 1.49e3T + 3.89e7T^{2} \)
83 \( 1 + (-6.73e3 + 6.73e3i)T - 4.74e7iT^{2} \)
89 \( 1 + (-2.52e3 - 2.52e3i)T + 6.27e7iT^{2} \)
97 \( 1 + (-3.47e3 + 3.47e3i)T - 8.85e7iT^{2} \)
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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

−19.09661880977548610836560836528, −17.92232156406487170401683007210, −17.30888391880964086051228324261, −15.94376057722748236601611084890, −14.82863745465711893161569532073, −12.16476027889271148958948763751, −11.18225639000555647234326714589, −8.965133156006368241761723611016, −7.30050391189314276652359369699, −5.52959134200637178627361099110, 0.71423285713753686510435011964, 5.15852910004743534924312568457, 7.84481413203184190526040178581, 10.06059520603840747602655506567, 11.18732214350939932892789591931, 12.09488134429474238145240711201, 14.37645387260727156379582149475, 16.45940416020549184690836829837, 17.53328923954207996685261283922, 18.32231062386337690514854231415

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