Properties

Label 2-65-65.29-c1-0-2
Degree $2$
Conductor $65$
Sign $0.995 - 0.0988i$
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  + (−2.20 + 1.27i)2-s + (1.86 − 1.07i)3-s + (2.24 − 3.88i)4-s + (−0.817 − 2.08i)5-s + (−2.74 + 4.74i)6-s + (2.54 + 1.46i)7-s + 6.31i·8-s + (0.817 − 1.41i)9-s + (4.45 + 3.54i)10-s + (0.317 + 0.550i)11-s − 9.64i·12-s + (−3.60 − 0.0716i)13-s − 7.48·14-s + (−3.76 − 3.00i)15-s + (−3.55 − 6.16i)16-s + (1.05 + 0.611i)17-s + ⋯
L(s)  = 1  + (−1.55 + 0.900i)2-s + (1.07 − 0.621i)3-s + (1.12 − 1.94i)4-s + (−0.365 − 0.930i)5-s + (−1.11 + 1.93i)6-s + (0.961 + 0.555i)7-s + 2.23i·8-s + (0.272 − 0.472i)9-s + (1.40 + 1.12i)10-s + (0.0957 + 0.165i)11-s − 2.78i·12-s + (−0.999 − 0.0198i)13-s − 1.99·14-s + (−0.972 − 0.774i)15-s + (−0.889 − 1.54i)16-s + (0.257 + 0.148i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.995 - 0.0988i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1/2),\ 0.995 - 0.0988i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.615744 + 0.0305131i\)
\(L(\frac12)\) \(\approx\) \(0.615744 + 0.0305131i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.817 + 2.08i)T \)
13 \( 1 + (3.60 + 0.0716i)T \)
good2 \( 1 + (2.20 - 1.27i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-1.86 + 1.07i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (-2.54 - 1.46i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.317 - 0.550i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.05 - 0.611i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.682 + 1.18i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.86 - 1.07i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 8.96T + 31T^{2} \)
37 \( 1 + (-1.05 + 0.611i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.98 - 8.62i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.18 + 0.683i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 6.16iT - 47T^{2} \)
53 \( 1 + 0.642iT - 53T^{2} \)
59 \( 1 + (-3.79 + 6.57i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.13 + 1.96i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.95 + 4.01i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.31 - 2.28i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 10.3iT - 73T^{2} \)
79 \( 1 + 1.03T + 79T^{2} \)
83 \( 1 + 11.8iT - 83T^{2} \)
89 \( 1 + (6.27 + 10.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-12.8 - 7.39i)T + (48.5 + 84.0i)T^{2} \)
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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

−14.98022890676848131633345210985, −14.39801196035490151699074589955, −12.75367365698710788691885012850, −11.38482144849478288861816217983, −9.653689505928242933990662182662, −8.741675433503530278349583814273, −8.036798912207701986494887292028, −7.26730989242016054952155691432, −5.28589253151984631524995208453, −1.82055882492538526323385586890, 2.46051051223476293834244925197, 3.84729037660810411257783261096, 7.37850850880326461968919535630, 8.069108541252817110607777368970, 9.284845587484303900667959922794, 10.22476348133857371197587830702, 11.06104589170493826314093471902, 12.11453446263063034029699268833, 14.12486954776046143906245384991, 14.83378457594605434614197342754

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