Properties

Label 2-13-13.11-c4-0-3
Degree $2$
Conductor $13$
Sign $0.201 + 0.979i$
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  + (0.540 − 2.01i)2-s + (−8.21 − 14.2i)3-s + (10.0 + 5.82i)4-s + (17.2 + 17.2i)5-s + (−33.1 + 8.87i)6-s + (−2.95 − 11.0i)7-s + (40.8 − 40.8i)8-s + (−94.3 + 163. i)9-s + (44.1 − 25.4i)10-s + (101. + 27.2i)11-s − 191. i·12-s + (−166. + 27.5i)13-s − 23.7·14-s + (103. − 387. i)15-s + (32.8 + 56.9i)16-s + (65.7 + 37.9i)17-s + ⋯
L(s)  = 1  + (0.135 − 0.504i)2-s + (−0.912 − 1.58i)3-s + (0.630 + 0.363i)4-s + (0.690 + 0.690i)5-s + (−0.920 + 0.246i)6-s + (−0.0602 − 0.224i)7-s + (0.637 − 0.637i)8-s + (−1.16 + 2.01i)9-s + (0.441 − 0.254i)10-s + (0.840 + 0.225i)11-s − 1.32i·12-s + (−0.986 + 0.163i)13-s − 0.121·14-s + (0.460 − 1.72i)15-s + (0.128 + 0.222i)16-s + (0.227 + 0.131i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.201 + 0.979i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.201 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13\)
Sign: $0.201 + 0.979i$
Analytic conductor: \(1.34380\)
Root analytic conductor: \(1.15922\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{13} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 13,\ (\ :2),\ 0.201 + 0.979i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.874347 - 0.713094i\)
\(L(\frac12)\) \(\approx\) \(0.874347 - 0.713094i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (166. - 27.5i)T \)
good2 \( 1 + (-0.540 + 2.01i)T + (-13.8 - 8i)T^{2} \)
3 \( 1 + (8.21 + 14.2i)T + (-40.5 + 70.1i)T^{2} \)
5 \( 1 + (-17.2 - 17.2i)T + 625iT^{2} \)
7 \( 1 + (2.95 + 11.0i)T + (-2.07e3 + 1.20e3i)T^{2} \)
11 \( 1 + (-101. - 27.2i)T + (1.26e4 + 7.32e3i)T^{2} \)
17 \( 1 + (-65.7 - 37.9i)T + (4.17e4 + 7.23e4i)T^{2} \)
19 \( 1 + (313. - 84.0i)T + (1.12e5 - 6.51e4i)T^{2} \)
23 \( 1 + (174. - 100. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (-432. - 748. i)T + (-3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (767. + 767. i)T + 9.23e5iT^{2} \)
37 \( 1 + (756. + 202. i)T + (1.62e6 + 9.37e5i)T^{2} \)
41 \( 1 + (-308. + 1.15e3i)T + (-2.44e6 - 1.41e6i)T^{2} \)
43 \( 1 + (-318. - 183. i)T + (1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (308. - 308. i)T - 4.87e6iT^{2} \)
53 \( 1 + 2.12e3T + 7.89e6T^{2} \)
59 \( 1 + (1.46e3 + 5.48e3i)T + (-1.04e7 + 6.05e6i)T^{2} \)
61 \( 1 + (2.85e3 - 4.93e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (525. - 1.96e3i)T + (-1.74e7 - 1.00e7i)T^{2} \)
71 \( 1 + (-1.51e3 + 404. i)T + (2.20e7 - 1.27e7i)T^{2} \)
73 \( 1 + (1.57e3 - 1.57e3i)T - 2.83e7iT^{2} \)
79 \( 1 - 4.44e3T + 3.89e7T^{2} \)
83 \( 1 + (4.02e3 + 4.02e3i)T + 4.74e7iT^{2} \)
89 \( 1 + (-1.06e4 - 2.84e3i)T + (5.43e7 + 3.13e7i)T^{2} \)
97 \( 1 + (-1.65e4 + 4.44e3i)T + (7.66e7 - 4.42e7i)T^{2} \)
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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

−18.90826986056104873850189907219, −17.56630083862672651920835367388, −16.77039680424137194139271730442, −14.24219869662379485214340282160, −12.78741550070334309428480591621, −11.87378830002541878809540646218, −10.56023016012891677312239992932, −7.34030654587919998860237464228, −6.34118575478990446880202059557, −2.00729610404056295788149045289, 4.89597969184485138270516817094, 6.13372198019449794045094056781, 9.307366467293753463033818686483, 10.55186080402038262078745605636, 11.98376219291088216615618937899, 14.48067049180093776696233150504, 15.57454680097728349239147092060, 16.73384681997925912619247158058, 17.25233112876021800161060602507, 19.83644591926455842809686550483

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