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Properties

Label 2-69-3.2-c6-0-37
Degree $2$
Conductor $69$
Sign $-0.999 + 0.0150i$
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  − 4.53i·2-s + (−0.406 − 26.9i)3-s + 43.4·4-s + 169. i·5-s + (−122. + 1.84i)6-s − 134.·7-s − 487. i·8-s + (−728. + 21.9i)9-s + 769.·10-s − 2.61e3i·11-s + (−17.6 − 1.17e3i)12-s − 3.44e3·13-s + 608. i·14-s + (4.58e3 − 69.0i)15-s + 572.·16-s − 3.82e3i·17-s + ⋯
L(s)  = 1  − 0.566i·2-s + (−0.0150 − 0.999i)3-s + 0.678·4-s + 1.35i·5-s + (−0.566 + 0.00853i)6-s − 0.391·7-s − 0.951i·8-s + (−0.999 + 0.0301i)9-s + 0.769·10-s − 1.96i·11-s + (−0.0102 − 0.678i)12-s − 1.56·13-s + 0.221i·14-s + (1.35 − 0.0204i)15-s + 0.139·16-s − 0.778i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0150i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.999 + 0.0150i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $-0.999 + 0.0150i$
Analytic conductor: \(15.8737\)
Root analytic conductor: \(3.98418\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :3),\ -0.999 + 0.0150i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.00894951 - 1.18763i\)
\(L(\frac12)\) \(\approx\) \(0.00894951 - 1.18763i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.406 + 26.9i)T \)
23 \( 1 - 2.53e3iT \)
good2 \( 1 + 4.53iT - 64T^{2} \)
5 \( 1 - 169. iT - 1.56e4T^{2} \)
7 \( 1 + 134.T + 1.17e5T^{2} \)
11 \( 1 + 2.61e3iT - 1.77e6T^{2} \)
13 \( 1 + 3.44e3T + 4.82e6T^{2} \)
17 \( 1 + 3.82e3iT - 2.41e7T^{2} \)
19 \( 1 + 6.53e3T + 4.70e7T^{2} \)
29 \( 1 + 2.96e4iT - 5.94e8T^{2} \)
31 \( 1 - 1.74e4T + 8.87e8T^{2} \)
37 \( 1 - 2.51e4T + 2.56e9T^{2} \)
41 \( 1 - 4.49e4iT - 4.75e9T^{2} \)
43 \( 1 - 346.T + 6.32e9T^{2} \)
47 \( 1 - 1.63e5iT - 1.07e10T^{2} \)
53 \( 1 + 1.90e5iT - 2.21e10T^{2} \)
59 \( 1 + 1.71e4iT - 4.21e10T^{2} \)
61 \( 1 - 3.01e5T + 5.15e10T^{2} \)
67 \( 1 - 2.34e4T + 9.04e10T^{2} \)
71 \( 1 + 1.47e5iT - 1.28e11T^{2} \)
73 \( 1 - 5.46e5T + 1.51e11T^{2} \)
79 \( 1 + 5.40e5T + 2.43e11T^{2} \)
83 \( 1 + 1.13e4iT - 3.26e11T^{2} \)
89 \( 1 + 5.30e5iT - 4.96e11T^{2} \)
97 \( 1 + 4.24e5T + 8.32e11T^{2} \)
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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

−12.84900150688781325155318046883, −11.59254606260949797656598042513, −11.10741151577436102078277464446, −9.842670761007635522134372568968, −7.947263460640570864085527147363, −6.83033019861414663613058594315, −6.08930412307906869253830222806, −3.11701699618685949253745167459, −2.45649711351668005752970239996, −0.41805328923002205388313543211, 2.18809631526416824535315653340, 4.44397497686506413742158215157, 5.28317197377103308374497069593, 6.92318489682377997279808547860, 8.309902621603809997434410314364, 9.516980715610628196596575295901, 10.42652418686992399854319437250, 12.09953071247810785684253897473, 12.65447700644586042559792090170, 14.66756836890002624227622341154

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