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Properties

Label 2-69-69.68-c5-0-32
Degree $2$
Conductor $69$
Sign $-0.995 - 0.0993i$
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  − 9.90i·2-s + (13.3 − 8.04i)3-s − 66.1·4-s + 91.0·5-s + (−79.7 − 132. i)6-s − 106. i·7-s + 337. i·8-s + (113. − 214. i)9-s − 902. i·10-s − 429.·11-s + (−882. + 532. i)12-s − 363.·13-s − 1.05e3·14-s + (1.21e3 − 733. i)15-s + 1.23e3·16-s + 1.77e3·17-s + ⋯
L(s)  = 1  − 1.75i·2-s + (0.856 − 0.516i)3-s − 2.06·4-s + 1.62·5-s + (−0.903 − 1.49i)6-s − 0.820i·7-s + 1.86i·8-s + (0.466 − 0.884i)9-s − 2.85i·10-s − 1.07·11-s + (−1.76 + 1.06i)12-s − 0.596·13-s − 1.43·14-s + (1.39 − 0.841i)15-s + 1.20·16-s + 1.49·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0993i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $-0.995 - 0.0993i$
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: \((2,\ 69,\ (\ :5/2),\ -0.995 - 0.0993i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.124375 + 2.49671i\)
\(L(\frac12)\) \(\approx\) \(0.124375 + 2.49671i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-13.3 + 8.04i)T \)
23 \( 1 + (-2.03e3 - 1.51e3i)T \)
good2 \( 1 + 9.90iT - 32T^{2} \)
5 \( 1 - 91.0T + 3.12e3T^{2} \)
7 \( 1 + 106. iT - 1.68e4T^{2} \)
11 \( 1 + 429.T + 1.61e5T^{2} \)
13 \( 1 + 363.T + 3.71e5T^{2} \)
17 \( 1 - 1.77e3T + 1.41e6T^{2} \)
19 \( 1 - 1.10e3iT - 2.47e6T^{2} \)
29 \( 1 - 3.05e3iT - 2.05e7T^{2} \)
31 \( 1 + 6.82e3T + 2.86e7T^{2} \)
37 \( 1 + 3.53e3iT - 6.93e7T^{2} \)
41 \( 1 + 1.55e4iT - 1.15e8T^{2} \)
43 \( 1 - 1.58e4iT - 1.47e8T^{2} \)
47 \( 1 - 1.84e3iT - 2.29e8T^{2} \)
53 \( 1 - 2.93e4T + 4.18e8T^{2} \)
59 \( 1 - 1.73e4iT - 7.14e8T^{2} \)
61 \( 1 + 2.52e4iT - 8.44e8T^{2} \)
67 \( 1 - 1.23e4iT - 1.35e9T^{2} \)
71 \( 1 + 8.45e4iT - 1.80e9T^{2} \)
73 \( 1 + 8.01e3T + 2.07e9T^{2} \)
79 \( 1 - 6.62e4iT - 3.07e9T^{2} \)
83 \( 1 - 4.50e4T + 3.93e9T^{2} \)
89 \( 1 + 2.39e4T + 5.58e9T^{2} \)
97 \( 1 + 9.67e4iT - 8.58e9T^{2} \)
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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

−13.07283365271996137306635253856, −12.39479269118120274928630907307, −10.60291471427379339302753427990, −9.966182045509712623689115392597, −9.119872376617698062812615585764, −7.48445313935786181213631416121, −5.36966866456018646294850588321, −3.41532604841663436318022480749, −2.23073144460425869929292385984, −1.13296086876584902883551651857, 2.55411265803536764271260568979, 5.06042611617143028832022002524, 5.67786179426324893052046888679, 7.24639340807839243748865269156, 8.490533472750072834875928589345, 9.394078278310876838015483737598, 10.20761226537221393515466372797, 12.90948845583354708189846176375, 13.65070604017211529685640159593, 14.62364374422108699317436110103

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