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Properties

Label 2-65-13.12-c1-0-1
Degree $2$
Conductor $65$
Sign $0.899 - 0.435i$
Analytic cond. $0.519027$
Root an. cond. $0.720435$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.571i·2-s − 0.428·3-s + 1.67·4-s − i·5-s − 0.244i·6-s + 2.67i·7-s + 2.10i·8-s − 2.81·9-s + 0.571·10-s − 5.10i·11-s − 0.715·12-s + (−1.57 − 3.24i)13-s − 1.52·14-s + 0.428i·15-s + 2.14·16-s − 5.34·17-s + ⋯
L(s)  = 1  + 0.404i·2-s − 0.247·3-s + 0.836·4-s − 0.447i·5-s − 0.0999i·6-s + 1.01i·7-s + 0.742i·8-s − 0.938·9-s + 0.180·10-s − 1.53i·11-s − 0.206·12-s + (−0.435 − 0.899i)13-s − 0.408·14-s + 0.110i·15-s + 0.535·16-s − 1.29·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.435i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.899 - 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.899 - 0.435i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1/2),\ 0.899 - 0.435i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.905227 + 0.207727i\)
\(L(\frac12)\) \(\approx\) \(0.905227 + 0.207727i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + iT \)
13 \( 1 + (1.57 + 3.24i)T \)
good2 \( 1 - 0.571iT - 2T^{2} \)
3 \( 1 + 0.428T + 3T^{2} \)
7 \( 1 - 2.67iT - 7T^{2} \)
11 \( 1 + 5.10iT - 11T^{2} \)
17 \( 1 + 5.34T + 17T^{2} \)
19 \( 1 - 6.24iT - 19T^{2} \)
23 \( 1 - 2.42T + 23T^{2} \)
29 \( 1 - 2.67T + 29T^{2} \)
31 \( 1 + 0.244iT - 31T^{2} \)
37 \( 1 + 3.32iT - 37T^{2} \)
41 \( 1 - 6.48iT - 41T^{2} \)
43 \( 1 - 10.9T + 43T^{2} \)
47 \( 1 + 2.67iT - 47T^{2} \)
53 \( 1 + 4.20T + 53T^{2} \)
59 \( 1 - 0.899iT - 59T^{2} \)
61 \( 1 + 5.81T + 61T^{2} \)
67 \( 1 - 2.18iT - 67T^{2} \)
71 \( 1 - 6.24iT - 71T^{2} \)
73 \( 1 + 10.9iT - 73T^{2} \)
79 \( 1 + 3.63T + 79T^{2} \)
83 \( 1 - 9.81iT - 83T^{2} \)
89 \( 1 + 7.63iT - 89T^{2} \)
97 \( 1 + 11.3iT - 97T^{2} \)
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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

−15.16460029849831355120241114897, −14.10342648637081388964377878437, −12.59743636928104245143390027488, −11.60837775717027520466664113516, −10.75746894082536747495287934991, −8.850415962085183974280220793406, −8.035550335690015358908252363190, −6.14585082942880469777021637917, −5.51102560512337133794403396456, −2.80712830311512006685651363026, 2.46972928728870805519330112331, 4.48298773849430596679237318785, 6.65910692770411291565565913649, 7.26603661263541042190303201899, 9.319945487646810336757114859379, 10.63542131962754570323670551521, 11.29787395438233315157899428719, 12.37782784925945569524643485038, 13.71351142861903819214329729372, 14.88002925724217395648853301900

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