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Properties

Label 2-65-65.8-c1-0-3
Degree $2$
Conductor $65$
Sign $0.962 + 0.272i$
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  + 2.03·2-s + (−1.32 − 1.32i)3-s + 2.12·4-s + (−1.45 + 1.70i)5-s + (−2.70 − 2.70i)6-s + 1.61i·7-s + 0.248·8-s + 0.537i·9-s + (−2.94 + 3.45i)10-s + (2.70 − 2.70i)11-s + (−2.82 − 2.82i)12-s + (−1.04 + 3.45i)13-s + 3.28i·14-s + (4.19 − 0.329i)15-s − 3.74·16-s + (−2.24 − 2.24i)17-s + ⋯
L(s)  = 1  + 1.43·2-s + (−0.767 − 0.767i)3-s + 1.06·4-s + (−0.649 + 0.760i)5-s + (−1.10 − 1.10i)6-s + 0.611i·7-s + 0.0877·8-s + 0.179i·9-s + (−0.932 + 1.09i)10-s + (0.814 − 0.814i)11-s + (−0.814 − 0.814i)12-s + (−0.288 + 0.957i)13-s + 0.878i·14-s + (1.08 − 0.0852i)15-s − 0.935·16-s + (−0.545 − 0.545i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.272i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 + 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.962 + 0.272i$
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: \((2,\ 65,\ (\ :1/2),\ 0.962 + 0.272i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27201 - 0.176875i\)
\(L(\frac12)\) \(\approx\) \(1.27201 - 0.176875i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.45 - 1.70i)T \)
13 \( 1 + (1.04 - 3.45i)T \)
good2 \( 1 - 2.03T + 2T^{2} \)
3 \( 1 + (1.32 + 1.32i)T + 3iT^{2} \)
7 \( 1 - 1.61iT - 7T^{2} \)
11 \( 1 + (-2.70 + 2.70i)T - 11iT^{2} \)
17 \( 1 + (2.24 + 2.24i)T + 17iT^{2} \)
19 \( 1 + (-2.32 + 2.32i)T - 19iT^{2} \)
23 \( 1 + (-4.82 + 4.82i)T - 23iT^{2} \)
29 \( 1 - 4.27iT - 29T^{2} \)
31 \( 1 + (-3.36 - 3.36i)T + 31iT^{2} \)
37 \( 1 - 7.78iT - 37T^{2} \)
41 \( 1 + (-2.87 - 2.87i)T + 41iT^{2} \)
43 \( 1 + (3.97 - 3.97i)T - 43iT^{2} \)
47 \( 1 + 5.36iT - 47T^{2} \)
53 \( 1 + (4.61 + 4.61i)T + 53iT^{2} \)
59 \( 1 + (4.47 + 4.47i)T + 59iT^{2} \)
61 \( 1 + 12.1T + 61T^{2} \)
67 \( 1 - 5.84T + 67T^{2} \)
71 \( 1 + (1.37 + 1.37i)T + 71iT^{2} \)
73 \( 1 - 4.02T + 73T^{2} \)
79 \( 1 - 8.63iT - 79T^{2} \)
83 \( 1 + 7.48iT - 83T^{2} \)
89 \( 1 + (8.59 + 8.59i)T + 89iT^{2} \)
97 \( 1 - 8.63T + 97T^{2} \)
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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

−14.62308420269136391512892241404, −13.76883522291476925077404576273, −12.53606979389199727396909673327, −11.68197347988629349298891547751, −11.27480830383269298879905578546, −8.957737779680520088708344110804, −6.85788341600183512872207704174, −6.38620527392712089278701628847, −4.79762587947333174023710871602, −3.09654214130448017941086570478, 3.87176864324593968150339037411, 4.68820107006109882557049776490, 5.77227257833640586987433775080, 7.50861317309983544923769073311, 9.426359643764365961933374898690, 10.85935123902644845756961988026, 11.85698316280401274860908836129, 12.70230055386846776747464588509, 13.74901819567257151969081898012, 15.16139806512324215239960689958

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