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Properties

Label 2-69-3.2-c6-0-12
Degree $2$
Conductor $69$
Sign $0.156 + 0.987i$
Analytic cond. $15.8737$
Root an. cond. $3.98418$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 12.3i·2-s + (−26.6 + 4.21i)3-s − 89.2·4-s + 195. i·5-s + (52.2 + 330. i)6-s − 657.·7-s + 312. i·8-s + (693. − 225. i)9-s + 2.42e3·10-s − 1.22e3i·11-s + (2.37e3 − 376. i)12-s + 1.91e3·13-s + 8.14e3i·14-s + (−825. − 5.21e3i)15-s − 1.84e3·16-s − 1.07e3i·17-s + ⋯
L(s)  = 1  − 1.54i·2-s + (−0.987 + 0.156i)3-s − 1.39·4-s + 1.56i·5-s + (0.241 + 1.52i)6-s − 1.91·7-s + 0.609i·8-s + (0.951 − 0.308i)9-s + 2.42·10-s − 0.923i·11-s + (1.37 − 0.217i)12-s + 0.872·13-s + 2.96i·14-s + (−0.244 − 1.54i)15-s − 0.451·16-s − 0.217i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $0.156 + 0.987i$
Analytic conductor: \(15.8737\)
Root analytic conductor: \(3.98418\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :3),\ 0.156 + 0.987i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.657534 - 0.561667i\)
\(L(\frac12)\) \(\approx\) \(0.657534 - 0.561667i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (26.6 - 4.21i)T \)
23 \( 1 - 2.53e3iT \)
good2 \( 1 + 12.3iT - 64T^{2} \)
5 \( 1 - 195. iT - 1.56e4T^{2} \)
7 \( 1 + 657.T + 1.17e5T^{2} \)
11 \( 1 + 1.22e3iT - 1.77e6T^{2} \)
13 \( 1 - 1.91e3T + 4.82e6T^{2} \)
17 \( 1 + 1.07e3iT - 2.41e7T^{2} \)
19 \( 1 - 9.68e3T + 4.70e7T^{2} \)
29 \( 1 - 9.39e3iT - 5.94e8T^{2} \)
31 \( 1 - 2.35e4T + 8.87e8T^{2} \)
37 \( 1 - 5.20e4T + 2.56e9T^{2} \)
41 \( 1 - 4.19e4iT - 4.75e9T^{2} \)
43 \( 1 - 8.88e3T + 6.32e9T^{2} \)
47 \( 1 + 8.08e4iT - 1.07e10T^{2} \)
53 \( 1 - 4.64e4iT - 2.21e10T^{2} \)
59 \( 1 + 3.52e5iT - 4.21e10T^{2} \)
61 \( 1 - 7.14e4T + 5.15e10T^{2} \)
67 \( 1 - 8.94e4T + 9.04e10T^{2} \)
71 \( 1 - 2.86e5iT - 1.28e11T^{2} \)
73 \( 1 + 4.50e5T + 1.51e11T^{2} \)
79 \( 1 - 2.76e5T + 2.43e11T^{2} \)
83 \( 1 + 6.07e5iT - 3.26e11T^{2} \)
89 \( 1 - 1.20e6iT - 4.96e11T^{2} \)
97 \( 1 - 1.12e6T + 8.32e11T^{2} \)
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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

−13.02137488010466169685033338113, −11.77529556468560345201982861845, −11.06474836244132065686744401379, −10.19088841921857530368768793991, −9.511747169516641677493167563585, −6.88930239178129363344519318818, −6.01777968864487951479241114235, −3.60479665455121512384465827522, −2.99371508876743204453409114012, −0.67592495849194144485160988516, 0.72646573371698831344097044552, 4.34157635615239998988017213004, 5.57577387777542388078182267168, 6.38310957643895499742213897716, 7.54549607236956185690524774579, 9.038622467786884865795097069311, 9.899047566691657894535456108048, 11.97710607630060324799888782601, 12.92583634818885591524558257023, 13.51715376160660978977948244733

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