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Properties

Label 2-65-65.8-c1-0-0
Degree $2$
Conductor $65$
Sign $0.828 - 0.559i$
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  − 1.57·2-s + (0.725 + 0.725i)3-s + 0.494·4-s + (2.23 + 0.146i)5-s + (−1.14 − 1.14i)6-s + 4.24i·7-s + 2.37·8-s − 1.94i·9-s + (−3.52 − 0.231i)10-s + (1.14 − 1.14i)11-s + (0.358 + 0.358i)12-s + (−3.59 − 0.231i)13-s − 6.71i·14-s + (1.51 + 1.72i)15-s − 4.74·16-s + (−4.37 − 4.37i)17-s + ⋯
L(s)  = 1  − 1.11·2-s + (0.419 + 0.419i)3-s + 0.247·4-s + (0.997 + 0.0654i)5-s + (−0.468 − 0.468i)6-s + 1.60i·7-s + 0.840·8-s − 0.648i·9-s + (−1.11 − 0.0731i)10-s + (0.345 − 0.345i)11-s + (0.103 + 0.103i)12-s + (−0.997 − 0.0641i)13-s − 1.79i·14-s + (0.390 + 0.445i)15-s − 1.18·16-s + (−1.06 − 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.828 - 0.559i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.828 - 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.828 - 0.559i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1/2),\ 0.828 - 0.559i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.615642 + 0.188251i\)
\(L(\frac12)\) \(\approx\) \(0.615642 + 0.188251i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.23 - 0.146i)T \)
13 \( 1 + (3.59 + 0.231i)T \)
good2 \( 1 + 1.57T + 2T^{2} \)
3 \( 1 + (-0.725 - 0.725i)T + 3iT^{2} \)
7 \( 1 - 4.24iT - 7T^{2} \)
11 \( 1 + (-1.14 + 1.14i)T - 11iT^{2} \)
17 \( 1 + (4.37 + 4.37i)T + 17iT^{2} \)
19 \( 1 + (-0.274 + 0.274i)T - 19iT^{2} \)
23 \( 1 + (-1.64 + 1.64i)T - 23iT^{2} \)
29 \( 1 - 2.79iT - 29T^{2} \)
31 \( 1 + (2.30 + 2.30i)T + 31iT^{2} \)
37 \( 1 - 2.04iT - 37T^{2} \)
41 \( 1 + (0.883 + 0.883i)T + 41iT^{2} \)
43 \( 1 + (0.944 - 0.944i)T - 43iT^{2} \)
47 \( 1 + 0.483iT - 47T^{2} \)
53 \( 1 + (7.24 + 7.24i)T + 53iT^{2} \)
59 \( 1 + (0.311 + 0.311i)T + 59iT^{2} \)
61 \( 1 - 11.2T + 61T^{2} \)
67 \( 1 + 7.11T + 67T^{2} \)
71 \( 1 + (-7.42 - 7.42i)T + 71iT^{2} \)
73 \( 1 + 4.96T + 73T^{2} \)
79 \( 1 - 10.7iT - 79T^{2} \)
83 \( 1 - 11.8iT - 83T^{2} \)
89 \( 1 + (-1.10 - 1.10i)T + 89iT^{2} \)
97 \( 1 - 10.7T + 97T^{2} \)
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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

−15.07934123419110404166072126474, −14.13699265301733970036242748481, −12.77177267987856534062652673417, −11.41234557789011245384755963616, −9.857427334127439545311226971635, −9.236354439295179674587682375078, −8.618386762246454991858458381969, −6.70901824852370365272094294555, −5.08808158779083335096813891818, −2.49041612812083156781982874075, 1.76599159402419681511857234190, 4.58971945404141182116384526896, 6.87648173898165201033931885842, 7.76299020334468258498450671409, 9.068373180837124512311948372167, 10.14696875298122029647479453950, 10.81249526149253172515314794159, 12.98882453638753997348013896174, 13.63190710284276713355609317445, 14.51540130742086983079971007852

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