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Properties

Label 2-69-23.22-c6-0-8
Degree $2$
Conductor $69$
Sign $0.988 - 0.150i$
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  − 10.6·2-s + 15.5·3-s + 49.0·4-s − 66.6i·5-s − 165.·6-s + 306. i·7-s + 159.·8-s + 243·9-s + 708. i·10-s − 998. i·11-s + 763.·12-s − 1.93e3·13-s − 3.26e3i·14-s − 1.03e3i·15-s − 4.83e3·16-s + 9.46e3i·17-s + ⋯
L(s)  = 1  − 1.32·2-s + 0.577·3-s + 0.765·4-s − 0.532i·5-s − 0.767·6-s + 0.894i·7-s + 0.311·8-s + 0.333·9-s + 0.708i·10-s − 0.750i·11-s + 0.442·12-s − 0.878·13-s − 1.18i·14-s − 0.307i·15-s − 1.17·16-s + 1.92i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.150i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.988 - 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $0.988 - 0.150i$
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: \((2,\ 69,\ (\ :3),\ 0.988 - 0.150i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.05871 + 0.0803572i\)
\(L(\frac12)\) \(\approx\) \(1.05871 + 0.0803572i\)
\(L(4)\) 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 \( 1 + (-1.20e4 + 1.83e3i)T \)
good2 \( 1 + 10.6T + 64T^{2} \)
5 \( 1 + 66.6iT - 1.56e4T^{2} \)
7 \( 1 - 306. iT - 1.17e5T^{2} \)
11 \( 1 + 998. iT - 1.77e6T^{2} \)
13 \( 1 + 1.93e3T + 4.82e6T^{2} \)
17 \( 1 - 9.46e3iT - 2.41e7T^{2} \)
19 \( 1 + 9.74e3iT - 4.70e7T^{2} \)
29 \( 1 - 4.41e4T + 5.94e8T^{2} \)
31 \( 1 - 1.29e4T + 8.87e8T^{2} \)
37 \( 1 - 9.38e4iT - 2.56e9T^{2} \)
41 \( 1 - 3.57e4T + 4.75e9T^{2} \)
43 \( 1 - 674. iT - 6.32e9T^{2} \)
47 \( 1 - 1.02e5T + 1.07e10T^{2} \)
53 \( 1 + 1.85e5iT - 2.21e10T^{2} \)
59 \( 1 - 7.62e4T + 4.21e10T^{2} \)
61 \( 1 + 4.05e5iT - 5.15e10T^{2} \)
67 \( 1 - 4.14e5iT - 9.04e10T^{2} \)
71 \( 1 - 1.12e5T + 1.28e11T^{2} \)
73 \( 1 + 1.20e5T + 1.51e11T^{2} \)
79 \( 1 - 3.85e5iT - 2.43e11T^{2} \)
83 \( 1 + 1.30e5iT - 3.26e11T^{2} \)
89 \( 1 - 1.03e6iT - 4.96e11T^{2} \)
97 \( 1 + 2.70e4iT - 8.32e11T^{2} \)
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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

−13.46290499151047971049563520076, −12.39071571152014017407570220172, −10.95323285617947536396056875303, −9.804082750856280055050963109713, −8.635967943695877288620385147797, −8.392483012077473046625492622510, −6.71390289165893979999202600980, −4.81150744421655264522041230150, −2.60607834076209406078136694916, −0.974950745172253054293332239307, 0.855303005707580462156772077479, 2.60835743142519008305208754511, 4.55332306312767648162996436551, 7.13962423537788334431371705108, 7.50022736327293254234758644102, 9.018213491213237636431974686304, 9.962050966594534204616921732233, 10.67678533314777869851374458673, 12.20790392435942459101909301884, 13.77253427364076622959205548517

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