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Properties

Label 2-65-65.63-c1-0-3
Degree $2$
Conductor $65$
Sign $-0.0753 + 0.997i$
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  + (−0.792 − 1.37i)2-s + (−0.0510 + 0.190i)3-s + (−0.255 + 0.442i)4-s + (0.0672 − 2.23i)5-s + (0.302 − 0.0809i)6-s + (−0.474 − 0.274i)7-s − 2.35·8-s + (2.56 + 1.48i)9-s + (−3.12 + 1.67i)10-s + (0.147 + 0.0396i)11-s + (−0.0713 − 0.0713i)12-s + (1.63 + 3.21i)13-s + 0.868i·14-s + (0.422 + 0.126i)15-s + (2.38 + 4.12i)16-s + (3.03 − 0.813i)17-s + ⋯
L(s)  = 1  + (−0.560 − 0.970i)2-s + (−0.0294 + 0.110i)3-s + (−0.127 + 0.221i)4-s + (0.0300 − 0.999i)5-s + (0.123 − 0.0330i)6-s + (−0.179 − 0.103i)7-s − 0.834·8-s + (0.854 + 0.493i)9-s + (−0.986 + 0.530i)10-s + (0.0446 + 0.0119i)11-s + (−0.0205 − 0.0205i)12-s + (0.452 + 0.891i)13-s + 0.232i·14-s + (0.109 + 0.0327i)15-s + (0.595 + 1.03i)16-s + (0.736 − 0.197i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0753 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0753 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $-0.0753 + 0.997i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1/2),\ -0.0753 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.485016 - 0.523070i\)
\(L(\frac12)\) \(\approx\) \(0.485016 - 0.523070i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.0672 + 2.23i)T \)
13 \( 1 + (-1.63 - 3.21i)T \)
good2 \( 1 + (0.792 + 1.37i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.0510 - 0.190i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (0.474 + 0.274i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.147 - 0.0396i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-3.03 + 0.813i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.18 - 4.40i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (3.41 + 0.916i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (2.02 - 1.17i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (6.60 + 6.60i)T + 31iT^{2} \)
37 \( 1 + (-5.89 + 3.40i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.926 + 3.45i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-1.84 - 6.86i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 - 9.13iT - 47T^{2} \)
53 \( 1 + (3.70 + 3.70i)T + 53iT^{2} \)
59 \( 1 + (-3.67 + 0.985i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (3.92 - 6.79i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.44 + 4.23i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (15.1 - 4.04i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 - 3.91T + 73T^{2} \)
79 \( 1 + 11.1iT - 79T^{2} \)
83 \( 1 - 13.4iT - 83T^{2} \)
89 \( 1 + (2.35 - 8.78i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-3.78 + 6.55i)T + (-48.5 - 84.0i)T^{2} \)
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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

−14.54010827587472084368143215543, −13.13672125378405602068028056112, −12.25101456205553704560598902475, −11.19669941377091881099800146852, −9.936650388884754974087346833230, −9.256699810708096623181939156054, −7.81957001443881392017076218921, −5.84761028848822928500607700236, −4.05106344134997214030815244768, −1.64534663088111388382263783234, 3.32474998237700577061427562947, 5.87674071619967259957876432714, 6.94843260491310952290373597392, 7.82960019219958791676129575329, 9.305789035240785137350314010954, 10.42755502092316803549311192649, 11.85355480910413215461018337072, 13.07542656566671152710898453065, 14.52815766831509508784964217239, 15.39168535283021975971631836571

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