Properties

Label 2-69-3.2-c6-0-25
Degree $2$
Conductor $69$
Sign $0.994 + 0.101i$
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  + 14.6i·2-s + (2.75 − 26.8i)3-s − 149.·4-s + 100. i·5-s + (392. + 40.2i)6-s − 263.·7-s − 1.25e3i·8-s + (−713. − 147. i)9-s − 1.47e3·10-s − 497. i·11-s + (−412. + 4.02e3i)12-s + 2.89e3·13-s − 3.84e3i·14-s + (2.70e3 + 277. i)15-s + 8.79e3·16-s − 8.32e3i·17-s + ⋯
L(s)  = 1  + 1.82i·2-s + (0.101 − 0.994i)3-s − 2.34·4-s + 0.806i·5-s + (1.81 + 0.186i)6-s − 0.767·7-s − 2.45i·8-s + (−0.979 − 0.202i)9-s − 1.47·10-s − 0.373i·11-s + (−0.238 + 2.33i)12-s + 1.31·13-s − 1.40i·14-s + (0.802 + 0.0821i)15-s + 2.14·16-s − 1.69i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.101i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.994 + 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $0.994 + 0.101i$
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: \((2,\ 69,\ (\ :3),\ 0.994 + 0.101i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.923893 - 0.0471854i\)
\(L(\frac12)\) \(\approx\) \(0.923893 - 0.0471854i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.75 + 26.8i)T \)
23 \( 1 + 2.53e3iT \)
good2 \( 1 - 14.6iT - 64T^{2} \)
5 \( 1 - 100. iT - 1.56e4T^{2} \)
7 \( 1 + 263.T + 1.17e5T^{2} \)
11 \( 1 + 497. iT - 1.77e6T^{2} \)
13 \( 1 - 2.89e3T + 4.82e6T^{2} \)
17 \( 1 + 8.32e3iT - 2.41e7T^{2} \)
19 \( 1 - 5.36e3T + 4.70e7T^{2} \)
29 \( 1 + 2.83e4iT - 5.94e8T^{2} \)
31 \( 1 + 3.76e4T + 8.87e8T^{2} \)
37 \( 1 + 4.32e4T + 2.56e9T^{2} \)
41 \( 1 - 5.16e4iT - 4.75e9T^{2} \)
43 \( 1 - 2.65e4T + 6.32e9T^{2} \)
47 \( 1 + 1.17e5iT - 1.07e10T^{2} \)
53 \( 1 + 1.15e5iT - 2.21e10T^{2} \)
59 \( 1 + 3.11e5iT - 4.21e10T^{2} \)
61 \( 1 + 3.70e4T + 5.15e10T^{2} \)
67 \( 1 - 4.84e5T + 9.04e10T^{2} \)
71 \( 1 - 1.53e5iT - 1.28e11T^{2} \)
73 \( 1 + 4.50e5T + 1.51e11T^{2} \)
79 \( 1 + 7.57e5T + 2.43e11T^{2} \)
83 \( 1 + 7.94e5iT - 3.26e11T^{2} \)
89 \( 1 - 4.54e5iT - 4.96e11T^{2} \)
97 \( 1 + 1.35e6T + 8.32e11T^{2} \)
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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

−13.77546642611360054761801103350, −13.06852971302365433022793121449, −11.39559127869922129651946082031, −9.505607696442552259488121033825, −8.391073974393162898825401969001, −7.20456959085165976458674217037, −6.56838431488630702150447807004, −5.54082773942434752187847106429, −3.29596918002072964881340722308, −0.39660828524867063776990960150, 1.38657096441042182120812482953, 3.28294998617259525316328216243, 4.13238288655832623951278903308, 5.54692074648124161119904774715, 8.613881434936337972899818136360, 9.225068254737938925111816938634, 10.34561054966100857434587894650, 11.06179418073849493175608341177, 12.40701180413020439374768718275, 13.07357554624768267321600559563

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