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Properties

Label 2-65-65.34-c2-0-0
Degree $2$
Conductor $65$
Sign $-0.985 - 0.167i$
Analytic cond. $1.77112$
Root an. cond. $1.33083$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.35 + 1.35i)2-s + 2.30i·3-s + 0.303i·4-s + (−4.94 − 0.749i)5-s + (−3.13 − 3.13i)6-s + (0.997 + 0.997i)7-s + (−5.85 − 5.85i)8-s + 3.67·9-s + (7.73 − 5.70i)10-s + (−11.2 + 11.2i)11-s − 0.700·12-s + (−12.5 + 3.54i)13-s − 2.71·14-s + (1.72 − 11.4i)15-s + 14.6·16-s + 20.4·17-s + ⋯
L(s)  = 1  + (−0.679 + 0.679i)2-s + 0.768i·3-s + 0.0758i·4-s + (−0.988 − 0.149i)5-s + (−0.522 − 0.522i)6-s + (0.142 + 0.142i)7-s + (−0.731 − 0.731i)8-s + 0.408·9-s + (0.773 − 0.570i)10-s + (−1.02 + 1.02i)11-s − 0.0583·12-s + (−0.962 + 0.272i)13-s − 0.193·14-s + (0.115 − 0.760i)15-s + 0.918·16-s + 1.20·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.167i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.985 - 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $-0.985 - 0.167i$
Analytic conductor: \(1.77112\)
Root analytic conductor: \(1.33083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1),\ -0.985 - 0.167i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0503521 + 0.596433i\)
\(L(\frac12)\) \(\approx\) \(0.0503521 + 0.596433i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (4.94 + 0.749i)T \)
13 \( 1 + (12.5 - 3.54i)T \)
good2 \( 1 + (1.35 - 1.35i)T - 4iT^{2} \)
3 \( 1 - 2.30iT - 9T^{2} \)
7 \( 1 + (-0.997 - 0.997i)T + 49iT^{2} \)
11 \( 1 + (11.2 - 11.2i)T - 121iT^{2} \)
17 \( 1 - 20.4T + 289T^{2} \)
19 \( 1 + (-26.2 - 26.2i)T + 361iT^{2} \)
23 \( 1 + 2.61T + 529T^{2} \)
29 \( 1 + 21.5T + 841T^{2} \)
31 \( 1 + (10.5 + 10.5i)T + 961iT^{2} \)
37 \( 1 + (-11.0 - 11.0i)T + 1.36e3iT^{2} \)
41 \( 1 + (13.2 + 13.2i)T + 1.68e3iT^{2} \)
43 \( 1 - 22.4T + 1.84e3T^{2} \)
47 \( 1 + (3.13 + 3.13i)T + 2.20e3iT^{2} \)
53 \( 1 - 46.2iT - 2.80e3T^{2} \)
59 \( 1 + (-20.1 + 20.1i)T - 3.48e3iT^{2} \)
61 \( 1 + 31.0T + 3.72e3T^{2} \)
67 \( 1 + (-64.5 + 64.5i)T - 4.48e3iT^{2} \)
71 \( 1 + (-64.3 - 64.3i)T + 5.04e3iT^{2} \)
73 \( 1 + (63.2 + 63.2i)T + 5.32e3iT^{2} \)
79 \( 1 - 46.7T + 6.24e3T^{2} \)
83 \( 1 + (-79.8 + 79.8i)T - 6.88e3iT^{2} \)
89 \( 1 + (2.66 - 2.66i)T - 7.92e3iT^{2} \)
97 \( 1 + (100. - 100. i)T - 9.40e3iT^{2} \)
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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

−15.42232696977684530065297987184, −14.69016129586145099990235595793, −12.64339829408202622755975794205, −11.95871404922076787920930467894, −10.18710437959808103893599036842, −9.451585427925319653849811443192, −7.80747503147088502307006141710, −7.41643453021259435781388802352, −5.14527877355187154847630872513, −3.64200081172527182718756226631, 0.69767218181431401430180437138, 2.91968289671929562253088330158, 5.32517821902837527762493680194, 7.29941978514506138487111475594, 8.100380368213764002989417356591, 9.643598346817507150466311529783, 10.82688672520575716532831679590, 11.71358534370168981813238101058, 12.71870843046529171172618094712, 14.03463087714682594608274220192

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