Properties

Label 2-69-23.12-c7-0-5
Degree $2$
Conductor $69$
Sign $-0.876 + 0.482i$
Analytic cond. $21.5545$
Root an. cond. $4.64268$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−13.0 + 8.38i)2-s + (−3.84 + 26.7i)3-s + (46.7 − 102. i)4-s + (2.38 − 0.700i)5-s + (−173. − 380. i)6-s + (540. + 623. i)7-s + (−34.3 − 238. i)8-s + (−699. − 205. i)9-s + (−25.2 + 29.1i)10-s + (3.78e3 + 2.43e3i)11-s + (2.55e3 + 1.64e3i)12-s + (−5.33e3 + 6.15e3i)13-s + (−1.22e4 − 3.60e3i)14-s + (9.55 + 66.4i)15-s + (1.18e4 + 1.37e4i)16-s + (1.39e4 + 3.05e4i)17-s + ⋯
L(s)  = 1  + (−1.15 + 0.740i)2-s + (−0.0821 + 0.571i)3-s + (0.364 − 0.799i)4-s + (0.00853 − 0.00250i)5-s + (−0.328 − 0.719i)6-s + (0.595 + 0.687i)7-s + (−0.0237 − 0.164i)8-s + (−0.319 − 0.0939i)9-s + (−0.00798 + 0.00921i)10-s + (0.857 + 0.551i)11-s + (0.426 + 0.274i)12-s + (−0.673 + 0.777i)13-s + (−1.19 − 0.351i)14-s + (0.000730 + 0.00508i)15-s + (0.724 + 0.836i)16-s + (0.689 + 1.50i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.876 + 0.482i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.876 + 0.482i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $-0.876 + 0.482i$
Analytic conductor: \(21.5545\)
Root analytic conductor: \(4.64268\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :7/2),\ -0.876 + 0.482i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.181056 - 0.704571i\)
\(L(\frac12)\) \(\approx\) \(0.181056 - 0.704571i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (3.84 - 26.7i)T \)
23 \( 1 + (-5.64e4 + 1.48e4i)T \)
good2 \( 1 + (13.0 - 8.38i)T + (53.1 - 116. i)T^{2} \)
5 \( 1 + (-2.38 + 0.700i)T + (6.57e4 - 4.22e4i)T^{2} \)
7 \( 1 + (-540. - 623. i)T + (-1.17e5 + 8.15e5i)T^{2} \)
11 \( 1 + (-3.78e3 - 2.43e3i)T + (8.09e6 + 1.77e7i)T^{2} \)
13 \( 1 + (5.33e3 - 6.15e3i)T + (-8.93e6 - 6.21e7i)T^{2} \)
17 \( 1 + (-1.39e4 - 3.05e4i)T + (-2.68e8 + 3.10e8i)T^{2} \)
19 \( 1 + (3.13e3 - 6.85e3i)T + (-5.85e8 - 6.75e8i)T^{2} \)
29 \( 1 + (-1.63e4 - 3.58e4i)T + (-1.12e10 + 1.30e10i)T^{2} \)
31 \( 1 + (3.04e4 + 2.11e5i)T + (-2.63e10 + 7.75e9i)T^{2} \)
37 \( 1 + (4.72e5 + 1.38e5i)T + (7.98e10 + 5.13e10i)T^{2} \)
41 \( 1 + (-4.82e5 + 1.41e5i)T + (1.63e11 - 1.05e11i)T^{2} \)
43 \( 1 + (3.81e4 - 2.65e5i)T + (-2.60e11 - 7.65e10i)T^{2} \)
47 \( 1 + 9.00e5T + 5.06e11T^{2} \)
53 \( 1 + (-4.93e4 - 5.69e4i)T + (-1.67e11 + 1.16e12i)T^{2} \)
59 \( 1 + (-1.62e6 + 1.87e6i)T + (-3.54e11 - 2.46e12i)T^{2} \)
61 \( 1 + (-4.62e5 - 3.22e6i)T + (-3.01e12 + 8.85e11i)T^{2} \)
67 \( 1 + (6.98e4 - 4.48e4i)T + (2.51e12 - 5.51e12i)T^{2} \)
71 \( 1 + (3.97e6 - 2.55e6i)T + (3.77e12 - 8.27e12i)T^{2} \)
73 \( 1 + (1.06e4 - 2.34e4i)T + (-7.23e12 - 8.34e12i)T^{2} \)
79 \( 1 + (3.54e6 - 4.09e6i)T + (-2.73e12 - 1.90e13i)T^{2} \)
83 \( 1 + (1.30e6 + 3.81e5i)T + (2.28e13 + 1.46e13i)T^{2} \)
89 \( 1 + (-1.02e6 + 7.11e6i)T + (-4.24e13 - 1.24e13i)T^{2} \)
97 \( 1 + (4.46e6 - 1.31e6i)T + (6.79e13 - 4.36e13i)T^{2} \)
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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

−14.65956123790769105108736871092, −12.63114441254006113865307766060, −11.48808748771120752693539697271, −10.10107496328235886267409230142, −9.225034220087053296540548146235, −8.326098816865737172497234751660, −7.05181592255856308610285823069, −5.70953854894830107609533495361, −4.03530539434644536901771300421, −1.65222108841592838265892325011, 0.41927523259504526923635785343, 1.37368814454130720421434035794, 2.99396997874669182507917647908, 5.18988512225720453377933558037, 7.10217619651621010118983528716, 8.085511376532652855609951743006, 9.256907465678007964705103759957, 10.38465502402351016158277439027, 11.40616872120184831322425927419, 12.17235296185996227935999674484

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