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Properties

Label 2-65-13.12-c5-0-7
Degree $2$
Conductor $65$
Sign $0.631 - 0.775i$
Analytic cond. $10.4249$
Root an. cond. $3.22876$
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  − 1.21i·2-s − 11.3·3-s + 30.5·4-s + 25i·5-s + 13.8i·6-s − 6.14i·7-s − 76.2i·8-s − 113.·9-s + 30.4·10-s + 391. i·11-s − 347.·12-s + (472. + 384. i)13-s − 7.48·14-s − 284. i·15-s + 883.·16-s + 1.54e3·17-s + ⋯
L(s)  = 1  − 0.215i·2-s − 0.729·3-s + 0.953·4-s + 0.447i·5-s + 0.157i·6-s − 0.0473i·7-s − 0.421i·8-s − 0.467·9-s + 0.0963·10-s + 0.974i·11-s − 0.695·12-s + (0.775 + 0.631i)13-s − 0.0102·14-s − 0.326i·15-s + 0.862·16-s + 1.29·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.631 - 0.775i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.631 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.631 - 0.775i$
Analytic conductor: \(10.4249\)
Root analytic conductor: \(3.22876\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :5/2),\ 0.631 - 0.775i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.40632 + 0.668348i\)
\(L(\frac12)\) \(\approx\) \(1.40632 + 0.668348i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - 25iT \)
13 \( 1 + (-472. - 384. i)T \)
good2 \( 1 + 1.21iT - 32T^{2} \)
3 \( 1 + 11.3T + 243T^{2} \)
7 \( 1 + 6.14iT - 1.68e4T^{2} \)
11 \( 1 - 391. iT - 1.61e5T^{2} \)
17 \( 1 - 1.54e3T + 1.41e6T^{2} \)
19 \( 1 - 3.01e3iT - 2.47e6T^{2} \)
23 \( 1 - 921.T + 6.43e6T^{2} \)
29 \( 1 + 5.24e3T + 2.05e7T^{2} \)
31 \( 1 - 1.74e3iT - 2.86e7T^{2} \)
37 \( 1 - 2.78e3iT - 6.93e7T^{2} \)
41 \( 1 - 7.30e3iT - 1.15e8T^{2} \)
43 \( 1 + 2.46e3T + 1.47e8T^{2} \)
47 \( 1 + 2.26e4iT - 2.29e8T^{2} \)
53 \( 1 + 2.73e4T + 4.18e8T^{2} \)
59 \( 1 + 4.76e4iT - 7.14e8T^{2} \)
61 \( 1 + 2.58e4T + 8.44e8T^{2} \)
67 \( 1 - 3.62e4iT - 1.35e9T^{2} \)
71 \( 1 - 2.52e4iT - 1.80e9T^{2} \)
73 \( 1 + 6.05e4iT - 2.07e9T^{2} \)
79 \( 1 - 8.76e4T + 3.07e9T^{2} \)
83 \( 1 - 8.24e4iT - 3.93e9T^{2} \)
89 \( 1 - 2.60e4iT - 5.58e9T^{2} \)
97 \( 1 + 1.35e5iT - 8.58e9T^{2} \)
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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

−14.23384085225799150989236604811, −12.48367334442932776369317706085, −11.76682285133220820503950232774, −10.81066140275556304489173877833, −9.860256434831619662488498393830, −7.894631987468914639427366702717, −6.64043896633120461430525062415, −5.61905341884784790673032839397, −3.49221941169252596560477264163, −1.64100649405658500953760186325, 0.820953959190761149791128661615, 3.05491424202200746819560979093, 5.37670275417430131767769711339, 6.15492354417010067842556343743, 7.65895551500918884874234207928, 8.946549399489030485128860341445, 10.79120380616896912831256556840, 11.31900200490332457983006735704, 12.42708618866790674053924275693, 13.70434539706976184419368830902

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