Properties

Label 2-69-23.22-c6-0-18
Degree $2$
Conductor $69$
Sign $-0.895 + 0.444i$
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  − 15.4·2-s + 15.5·3-s + 173.·4-s − 161. i·5-s − 240.·6-s − 585. i·7-s − 1.68e3·8-s + 243·9-s + 2.48e3i·10-s − 2.12e3i·11-s + 2.70e3·12-s + 1.92e3·13-s + 9.02e3i·14-s − 2.51e3i·15-s + 1.48e4·16-s + 1.37e3i·17-s + ⋯
L(s)  = 1  − 1.92·2-s + 0.577·3-s + 2.70·4-s − 1.29i·5-s − 1.11·6-s − 1.70i·7-s − 3.29·8-s + 0.333·9-s + 2.48i·10-s − 1.59i·11-s + 1.56·12-s + 0.877·13-s + 3.28i·14-s − 0.745i·15-s + 3.62·16-s + 0.279i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $-0.895 + 0.444i$
Analytic conductor: \(15.8737\)
Root analytic conductor: \(3.98418\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :3),\ -0.895 + 0.444i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.186958 - 0.797897i\)
\(L(\frac12)\) \(\approx\) \(0.186958 - 0.797897i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 15.5T \)
23 \( 1 + (1.09e4 - 5.40e3i)T \)
good2 \( 1 + 15.4T + 64T^{2} \)
5 \( 1 + 161. iT - 1.56e4T^{2} \)
7 \( 1 + 585. iT - 1.17e5T^{2} \)
11 \( 1 + 2.12e3iT - 1.77e6T^{2} \)
13 \( 1 - 1.92e3T + 4.82e6T^{2} \)
17 \( 1 - 1.37e3iT - 2.41e7T^{2} \)
19 \( 1 - 3.65e3iT - 4.70e7T^{2} \)
29 \( 1 - 6.03e3T + 5.94e8T^{2} \)
31 \( 1 - 3.25e3T + 8.87e8T^{2} \)
37 \( 1 + 2.55e4iT - 2.56e9T^{2} \)
41 \( 1 - 1.01e5T + 4.75e9T^{2} \)
43 \( 1 - 8.34e4iT - 6.32e9T^{2} \)
47 \( 1 + 1.19e5T + 1.07e10T^{2} \)
53 \( 1 + 8.70e4iT - 2.21e10T^{2} \)
59 \( 1 - 4.21e4T + 4.21e10T^{2} \)
61 \( 1 + 1.12e5iT - 5.15e10T^{2} \)
67 \( 1 + 4.98e4iT - 9.04e10T^{2} \)
71 \( 1 - 2.34e5T + 1.28e11T^{2} \)
73 \( 1 + 7.52e4T + 1.51e11T^{2} \)
79 \( 1 + 5.39e5iT - 2.43e11T^{2} \)
83 \( 1 - 2.21e5iT - 3.26e11T^{2} \)
89 \( 1 - 1.34e6iT - 4.96e11T^{2} \)
97 \( 1 + 1.22e6iT - 8.32e11T^{2} \)
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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

−12.99974213129508375253808590770, −11.37407726256460666013506420233, −10.47572657588083669654001534275, −9.380594864509353837820063506476, −8.348302091249288149160693302715, −7.82072633889347048183404476009, −6.23741022730675300613208375820, −3.64239725016103805085960465465, −1.34826972373617754064105510775, −0.59971014002545584907832687671, 2.00338571873144218187945990322, 2.77561094875409027891634945706, 6.23580871758574956965741137388, 7.23207082190165384738451679175, 8.399940536197413800965153417901, 9.363263937748726245317949112854, 10.22597618182687667169246358921, 11.38325620631051225113738797094, 12.36831204092008798283776443026, 14.62417382162482281392977020841

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