Properties

Label 2-13-13.8-c4-0-2
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: $\chi_{13} (8, \cdot )$
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

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

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