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Properties

Label 2-69-69.68-c7-0-45
Degree $2$
Conductor $69$
Sign $-0.594 + 0.803i$
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  + 2.66i·2-s + (1.60 + 46.7i)3-s + 120.·4-s − 103.·5-s + (−124. + 4.25i)6-s − 1.07e3i·7-s + 662. i·8-s + (−2.18e3 + 149. i)9-s − 274. i·10-s − 5.38e3·11-s + (193. + 5.65e3i)12-s − 6.04e3·13-s + 2.85e3·14-s + (−165. − 4.81e3i)15-s + 1.37e4·16-s − 2.97e4·17-s + ⋯
L(s)  = 1  + 0.235i·2-s + (0.0342 + 0.999i)3-s + 0.944·4-s − 0.368·5-s + (−0.234 + 0.00804i)6-s − 1.18i·7-s + 0.457i·8-s + (−0.997 + 0.0684i)9-s − 0.0867i·10-s − 1.21·11-s + (0.0323 + 0.944i)12-s − 0.763·13-s + 0.278·14-s + (−0.0126 − 0.368i)15-s + 0.837·16-s − 1.46·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.594 + 0.803i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.594 + 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $-0.594 + 0.803i$
Analytic conductor: \(21.5545\)
Root analytic conductor: \(4.64268\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :7/2),\ -0.594 + 0.803i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.0266771 - 0.0529283i\)
\(L(\frac12)\) \(\approx\) \(0.0266771 - 0.0529283i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.60 - 46.7i)T \)
23 \( 1 + (-4.56e4 - 3.62e4i)T \)
good2 \( 1 - 2.66iT - 128T^{2} \)
5 \( 1 + 103.T + 7.81e4T^{2} \)
7 \( 1 + 1.07e3iT - 8.23e5T^{2} \)
11 \( 1 + 5.38e3T + 1.94e7T^{2} \)
13 \( 1 + 6.04e3T + 6.27e7T^{2} \)
17 \( 1 + 2.97e4T + 4.10e8T^{2} \)
19 \( 1 + 2.32e3iT - 8.93e8T^{2} \)
29 \( 1 + 7.17e4iT - 1.72e10T^{2} \)
31 \( 1 + 1.65e5T + 2.75e10T^{2} \)
37 \( 1 + 5.27e5iT - 9.49e10T^{2} \)
41 \( 1 - 7.01e4iT - 1.94e11T^{2} \)
43 \( 1 - 6.09e5iT - 2.71e11T^{2} \)
47 \( 1 - 1.63e5iT - 5.06e11T^{2} \)
53 \( 1 - 4.17e5T + 1.17e12T^{2} \)
59 \( 1 + 1.70e6iT - 2.48e12T^{2} \)
61 \( 1 + 9.55e5iT - 3.14e12T^{2} \)
67 \( 1 - 2.07e6iT - 6.06e12T^{2} \)
71 \( 1 - 3.95e6iT - 9.09e12T^{2} \)
73 \( 1 - 4.34e6T + 1.10e13T^{2} \)
79 \( 1 - 7.84e6iT - 1.92e13T^{2} \)
83 \( 1 + 4.74e6T + 2.71e13T^{2} \)
89 \( 1 + 8.02e6T + 4.42e13T^{2} \)
97 \( 1 - 2.02e6iT - 8.07e13T^{2} \)
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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

−12.96036421277319503298524123188, −11.27454615987927659646288069402, −10.82699969578850165701346341985, −9.673900867190149442683583388785, −8.001055684745582387939294065558, −7.05289545080397559847075263163, −5.38587521456906580664978543205, −4.02665604468973303898612127822, −2.51319567405881429772449190771, −0.01782814358406331594303752636, 2.03054945407101611301777520610, 2.81644173850639130581866417890, 5.35878148729322530485690871415, 6.64872135845547718404635441114, 7.69012082095119559598886010206, 8.851946908705450495906348191924, 10.63645604445096923586942762705, 11.68642684354111507255768029739, 12.41106283679529373366386095031, 13.32265823481374635960379939259

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