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Properties

Label 2-69-69.68-c5-0-17
Degree $2$
Conductor $69$
Sign $0.801 - 0.598i$
Analytic cond. $11.0664$
Root an. cond. $3.32663$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.57i·2-s + (−12.8 + 8.75i)3-s + 29.5·4-s + 37.1·5-s + (−13.7 − 20.2i)6-s − 122. i·7-s + 96.6i·8-s + (89.6 − 225. i)9-s + 58.4i·10-s + 321.·11-s + (−380. + 258. i)12-s − 207.·13-s + 192.·14-s + (−479. + 325. i)15-s + 793.·16-s + 1.64e3·17-s + ⋯
L(s)  = 1  + 0.277i·2-s + (−0.827 + 0.561i)3-s + 0.922·4-s + 0.665·5-s + (−0.156 − 0.229i)6-s − 0.942i·7-s + 0.534i·8-s + (0.368 − 0.929i)9-s + 0.184i·10-s + 0.801·11-s + (−0.763 + 0.518i)12-s − 0.340·13-s + 0.261·14-s + (−0.550 + 0.373i)15-s + 0.774·16-s + 1.37·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 - 0.598i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.801 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $0.801 - 0.598i$
Analytic conductor: \(11.0664\)
Root analytic conductor: \(3.32663\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :5/2),\ 0.801 - 0.598i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.87217 + 0.621919i\)
\(L(\frac12)\) \(\approx\) \(1.87217 + 0.621919i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (12.8 - 8.75i)T \)
23 \( 1 + (-2.53e3 + 114. i)T \)
good2 \( 1 - 1.57iT - 32T^{2} \)
5 \( 1 - 37.1T + 3.12e3T^{2} \)
7 \( 1 + 122. iT - 1.68e4T^{2} \)
11 \( 1 - 321.T + 1.61e5T^{2} \)
13 \( 1 + 207.T + 3.71e5T^{2} \)
17 \( 1 - 1.64e3T + 1.41e6T^{2} \)
19 \( 1 - 2.88e3iT - 2.47e6T^{2} \)
29 \( 1 + 2.44e3iT - 2.05e7T^{2} \)
31 \( 1 + 244.T + 2.86e7T^{2} \)
37 \( 1 + 9.93e3iT - 6.93e7T^{2} \)
41 \( 1 - 1.74e4iT - 1.15e8T^{2} \)
43 \( 1 + 3.35e3iT - 1.47e8T^{2} \)
47 \( 1 + 1.50e4iT - 2.29e8T^{2} \)
53 \( 1 + 7.91e3T + 4.18e8T^{2} \)
59 \( 1 + 1.13e4iT - 7.14e8T^{2} \)
61 \( 1 + 2.59e4iT - 8.44e8T^{2} \)
67 \( 1 + 3.69e4iT - 1.35e9T^{2} \)
71 \( 1 - 5.28e4iT - 1.80e9T^{2} \)
73 \( 1 + 7.18e4T + 2.07e9T^{2} \)
79 \( 1 - 4.33e4iT - 3.07e9T^{2} \)
83 \( 1 - 1.07e5T + 3.93e9T^{2} \)
89 \( 1 + 1.12e3T + 5.58e9T^{2} \)
97 \( 1 + 2.67e4iT - 8.58e9T^{2} \)
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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

−14.18509862942352499187353455842, −12.49804271744781678651409841424, −11.55821821948312214714163235709, −10.41881503315544351542197839863, −9.723064361413992759195426562312, −7.66994981520401064824369656423, −6.45653102874011109615547367421, −5.50407786939411325281237690539, −3.69915357373700703979230745493, −1.33206432112357111356092214367, 1.27893399997193357876330113620, 2.66595880611685887169020878511, 5.28702331187473747630531701625, 6.33079170569334315500524575570, 7.36067775554720037958740733370, 9.209748798088403698032457988499, 10.50460080084017646535399169015, 11.63556376222812134578164724703, 12.22577182953355286199127186755, 13.34355681568220850291656190288

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