Properties

Label 2-69-69.68-c5-0-13
Degree $2$
Conductor $69$
Sign $0.986 - 0.163i$
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  + 7.06i·2-s + (−15.4 − 1.75i)3-s − 17.9·4-s − 82.3·5-s + (12.3 − 109. i)6-s − 96.3i·7-s + 99.5i·8-s + (236. + 54.2i)9-s − 581. i·10-s + 87.8·11-s + (277. + 31.3i)12-s + 303.·13-s + 680.·14-s + (1.27e3 + 144. i)15-s − 1.27e3·16-s + 460.·17-s + ⋯
L(s)  = 1  + 1.24i·2-s + (−0.993 − 0.112i)3-s − 0.559·4-s − 1.47·5-s + (0.140 − 1.24i)6-s − 0.742i·7-s + 0.550i·8-s + (0.974 + 0.223i)9-s − 1.83i·10-s + 0.218·11-s + (0.555 + 0.0628i)12-s + 0.498·13-s + 0.927·14-s + (1.46 + 0.165i)15-s − 1.24·16-s + 0.386·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.163i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.986 - 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $0.986 - 0.163i$
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: \((2,\ 69,\ (\ :5/2),\ 0.986 - 0.163i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.766505 + 0.0630573i\)
\(L(\frac12)\) \(\approx\) \(0.766505 + 0.0630573i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (15.4 + 1.75i)T \)
23 \( 1 + (-2.44e3 + 693. i)T \)
good2 \( 1 - 7.06iT - 32T^{2} \)
5 \( 1 + 82.3T + 3.12e3T^{2} \)
7 \( 1 + 96.3iT - 1.68e4T^{2} \)
11 \( 1 - 87.8T + 1.61e5T^{2} \)
13 \( 1 - 303.T + 3.71e5T^{2} \)
17 \( 1 - 460.T + 1.41e6T^{2} \)
19 \( 1 + 1.56e3iT - 2.47e6T^{2} \)
29 \( 1 - 4.09e3iT - 2.05e7T^{2} \)
31 \( 1 - 1.86e3T + 2.86e7T^{2} \)
37 \( 1 + 1.18e4iT - 6.93e7T^{2} \)
41 \( 1 + 1.21e4iT - 1.15e8T^{2} \)
43 \( 1 + 2.24e4iT - 1.47e8T^{2} \)
47 \( 1 + 2.74e3iT - 2.29e8T^{2} \)
53 \( 1 + 1.15e4T + 4.18e8T^{2} \)
59 \( 1 + 3.99e4iT - 7.14e8T^{2} \)
61 \( 1 - 2.44e4iT - 8.44e8T^{2} \)
67 \( 1 - 4.17e4iT - 1.35e9T^{2} \)
71 \( 1 - 3.38e4iT - 1.80e9T^{2} \)
73 \( 1 - 6.36e4T + 2.07e9T^{2} \)
79 \( 1 - 4.16e4iT - 3.07e9T^{2} \)
83 \( 1 + 7.23e4T + 3.93e9T^{2} \)
89 \( 1 - 9.26e3T + 5.58e9T^{2} \)
97 \( 1 + 2.84e4iT - 8.58e9T^{2} \)
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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

−14.04199101397497883880544775623, −12.62922499128363811973839536894, −11.44375609787975141083985835602, −10.76063356169045018948808791918, −8.676934170213706982991406034351, −7.34687852976921279830052304693, −6.90519136940703332598744719200, −5.30658064244650702727520461899, −4.04356844983768289509769111858, −0.49467219593005663050569877142, 1.11606198376620570384242195417, 3.32750665400719665752396062895, 4.54513804222226550276173583497, 6.35447956792023599308444086746, 7.919000334857987590046942515281, 9.557676061084146956620880923444, 10.79762330055780257097447993841, 11.66636074599789773204781586106, 12.04870144417087756548266707678, 13.06015873042124710106510130590

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