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Properties

Label 2-69-3.2-c6-0-34
Degree $2$
Conductor $69$
Sign $-0.534 + 0.845i$
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  − 9.87i·2-s + (22.8 + 14.4i)3-s − 33.5·4-s − 220. i·5-s + (142. − 225. i)6-s + 473.·7-s − 300. i·8-s + (313. + 658. i)9-s − 2.17e3·10-s + 1.89e3i·11-s + (−766. − 483. i)12-s + 1.95e3·13-s − 4.67e3i·14-s + (3.17e3 − 5.03e3i)15-s − 5.11e3·16-s − 7.39e3i·17-s + ⋯
L(s)  = 1  − 1.23i·2-s + (0.845 + 0.534i)3-s − 0.524·4-s − 1.76i·5-s + (0.659 − 1.04i)6-s + 1.37·7-s − 0.587i·8-s + (0.429 + 0.903i)9-s − 2.17·10-s + 1.42i·11-s + (−0.443 − 0.280i)12-s + 0.888·13-s − 1.70i·14-s + (0.941 − 1.49i)15-s − 1.24·16-s − 1.50i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.534 + 0.845i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.534 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $-0.534 + 0.845i$
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: \((2,\ 69,\ (\ :3),\ -0.534 + 0.845i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.41766 - 2.57226i\)
\(L(\frac12)\) \(\approx\) \(1.41766 - 2.57226i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-22.8 - 14.4i)T \)
23 \( 1 + 2.53e3iT \)
good2 \( 1 + 9.87iT - 64T^{2} \)
5 \( 1 + 220. iT - 1.56e4T^{2} \)
7 \( 1 - 473.T + 1.17e5T^{2} \)
11 \( 1 - 1.89e3iT - 1.77e6T^{2} \)
13 \( 1 - 1.95e3T + 4.82e6T^{2} \)
17 \( 1 + 7.39e3iT - 2.41e7T^{2} \)
19 \( 1 + 6.14e3T + 4.70e7T^{2} \)
29 \( 1 + 7.96e3iT - 5.94e8T^{2} \)
31 \( 1 - 3.07e3T + 8.87e8T^{2} \)
37 \( 1 - 1.71e4T + 2.56e9T^{2} \)
41 \( 1 + 2.28e4iT - 4.75e9T^{2} \)
43 \( 1 + 3.57e3T + 6.32e9T^{2} \)
47 \( 1 - 1.83e5iT - 1.07e10T^{2} \)
53 \( 1 - 1.33e5iT - 2.21e10T^{2} \)
59 \( 1 - 1.53e5iT - 4.21e10T^{2} \)
61 \( 1 + 2.32e5T + 5.15e10T^{2} \)
67 \( 1 - 4.75e5T + 9.04e10T^{2} \)
71 \( 1 + 5.64e4iT - 1.28e11T^{2} \)
73 \( 1 - 1.70e5T + 1.51e11T^{2} \)
79 \( 1 - 2.90e5T + 2.43e11T^{2} \)
83 \( 1 + 1.21e5iT - 3.26e11T^{2} \)
89 \( 1 + 1.23e5iT - 4.96e11T^{2} \)
97 \( 1 - 2.22e5T + 8.32e11T^{2} \)
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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

−12.93472739769513484520177609266, −12.04629781822453465914679738469, −10.94296433851994159780031405848, −9.631895394637345657072124001769, −8.871799853428381102100938237658, −7.74002031316243439774801856219, −4.81111244150777496788778819368, −4.28288038912800810230743218280, −2.18916387686278971286278604465, −1.19308534049286438849086462583, 2.00468224902816487851401735007, 3.60037303618811100861897103278, 5.98287630995310784951557101153, 6.76092786233463140364770507740, 8.051095526864413091601236912205, 8.455908026388297007230901645258, 10.71798090680227763272516609827, 11.38377784391229311153703005096, 13.48865965649705436444721021385, 14.31197137927973690764707319430

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