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Properties

Label 2-65-65.12-c2-0-8
Degree $2$
Conductor $65$
Sign $0.699 + 0.714i$
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  + (0.456 − 0.456i)2-s + (3.37 − 3.37i)3-s + 3.58i·4-s + (0.779 − 4.93i)5-s − 3.08i·6-s + (−7.82 + 7.82i)7-s + (3.45 + 3.45i)8-s − 13.8i·9-s + (−1.89 − 2.60i)10-s − 7.57i·11-s + (12.1 + 12.1i)12-s + (4.80 + 12.0i)13-s + 7.13i·14-s + (−14.0 − 19.3i)15-s − 11.1·16-s + (1.84 + 1.84i)17-s + ⋯
L(s)  = 1  + (0.228 − 0.228i)2-s + (1.12 − 1.12i)3-s + 0.895i·4-s + (0.155 − 0.987i)5-s − 0.513i·6-s + (−1.11 + 1.11i)7-s + (0.432 + 0.432i)8-s − 1.53i·9-s + (−0.189 − 0.260i)10-s − 0.688i·11-s + (1.00 + 1.00i)12-s + (0.369 + 0.929i)13-s + 0.509i·14-s + (−0.936 − 1.28i)15-s − 0.698·16-s + (0.108 + 0.108i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.699 + 0.714i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.699 + 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.699 + 0.714i$
Analytic conductor: \(1.77112\)
Root analytic conductor: \(1.33083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1),\ 0.699 + 0.714i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.57488 - 0.661651i\)
\(L(\frac12)\) \(\approx\) \(1.57488 - 0.661651i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.779 + 4.93i)T \)
13 \( 1 + (-4.80 - 12.0i)T \)
good2 \( 1 + (-0.456 + 0.456i)T - 4iT^{2} \)
3 \( 1 + (-3.37 + 3.37i)T - 9iT^{2} \)
7 \( 1 + (7.82 - 7.82i)T - 49iT^{2} \)
11 \( 1 + 7.57iT - 121T^{2} \)
17 \( 1 + (-1.84 - 1.84i)T + 289iT^{2} \)
19 \( 1 - 7.21T + 361T^{2} \)
23 \( 1 + (21.0 - 21.0i)T - 529iT^{2} \)
29 \( 1 + 22.4iT - 841T^{2} \)
31 \( 1 - 5.43iT - 961T^{2} \)
37 \( 1 + (-11.3 + 11.3i)T - 1.36e3iT^{2} \)
41 \( 1 + 33.5iT - 1.68e3T^{2} \)
43 \( 1 + (-52.4 + 52.4i)T - 1.84e3iT^{2} \)
47 \( 1 + (-11.2 + 11.2i)T - 2.20e3iT^{2} \)
53 \( 1 + (0.512 - 0.512i)T - 2.80e3iT^{2} \)
59 \( 1 + 66.4T + 3.48e3T^{2} \)
61 \( 1 - 16.2T + 3.72e3T^{2} \)
67 \( 1 + (74.5 - 74.5i)T - 4.48e3iT^{2} \)
71 \( 1 + 86.4iT - 5.04e3T^{2} \)
73 \( 1 + (68.6 + 68.6i)T + 5.32e3iT^{2} \)
79 \( 1 - 42.6iT - 6.24e3T^{2} \)
83 \( 1 + (-94.0 - 94.0i)T + 6.88e3iT^{2} \)
89 \( 1 - 87.2T + 7.92e3T^{2} \)
97 \( 1 + (-24.9 + 24.9i)T - 9.40e3iT^{2} \)
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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

−13.85410999656249347039244519472, −13.41394936893337720973963340058, −12.41133464221930508650290617244, −11.89301027453909961811875394269, −9.233455991049730434404732371263, −8.719578438824417206468094440000, −7.59331751409954702708139571912, −6.02803998408540585481874511732, −3.62611424367918853595724694098, −2.21161524850289505713584143755, 3.05246527320919105943698117310, 4.32555356106498058603484782223, 6.24879829805631047668256609079, 7.56525398360618483416227993644, 9.474430838628768292800975374788, 10.16216129038083546937379412001, 10.68800562134245796151552532164, 13.13608828517261365398535858819, 14.09128983496246782502562975757, 14.70231484705650491374030099597

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