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Properties

Label 2-65-65.63-c1-0-0
Degree $2$
Conductor $65$
Sign $0.182 - 0.983i$
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  + (0.511 + 0.885i)2-s + (−0.721 + 2.69i)3-s + (0.477 − 0.826i)4-s + (−1.69 − 1.45i)5-s + (−2.75 + 0.737i)6-s + (0.834 + 0.481i)7-s + 3.02·8-s + (−4.12 − 2.38i)9-s + (0.423 − 2.24i)10-s + (1.60 + 0.430i)11-s + (1.88 + 1.88i)12-s + (1.82 − 3.11i)13-s + 0.985i·14-s + (5.14 − 3.51i)15-s + (0.590 + 1.02i)16-s + (−7.00 + 1.87i)17-s + ⋯
L(s)  = 1  + (0.361 + 0.626i)2-s + (−0.416 + 1.55i)3-s + (0.238 − 0.413i)4-s + (−0.758 − 0.651i)5-s + (−1.12 + 0.301i)6-s + (0.315 + 0.182i)7-s + 1.06·8-s + (−1.37 − 0.794i)9-s + (0.133 − 0.710i)10-s + (0.484 + 0.129i)11-s + (0.542 + 0.542i)12-s + (0.505 − 0.863i)13-s + 0.263i·14-s + (1.32 − 0.907i)15-s + (0.147 + 0.255i)16-s + (−1.69 + 0.455i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.182 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.182 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.182 - 0.983i$
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: \((2,\ 65,\ (\ :1/2),\ 0.182 - 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.735806 + 0.611731i\)
\(L(\frac12)\) \(\approx\) \(0.735806 + 0.611731i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.69 + 1.45i)T \)
13 \( 1 + (-1.82 + 3.11i)T \)
good2 \( 1 + (-0.511 - 0.885i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.721 - 2.69i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (-0.834 - 0.481i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.60 - 0.430i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (7.00 - 1.87i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.707 + 2.64i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (3.72 + 0.997i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-0.253 + 0.146i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.125 + 0.125i)T + 31iT^{2} \)
37 \( 1 + (-3.53 + 2.04i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.79 - 6.69i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-2.05 - 7.67i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 - 7.84iT - 47T^{2} \)
53 \( 1 + (1.99 + 1.99i)T + 53iT^{2} \)
59 \( 1 + (-4.87 + 1.30i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (1.04 - 1.80i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.64 + 6.32i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-12.6 + 3.37i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 - 3.22T + 73T^{2} \)
79 \( 1 - 13.5iT - 79T^{2} \)
83 \( 1 + 8.56iT - 83T^{2} \)
89 \( 1 + (-0.134 + 0.500i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (3.75 - 6.50i)T + (-48.5 - 84.0i)T^{2} \)
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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

−15.40787238154428941756695205213, −14.64427349532378083676889388164, −13.07506489181365882457875975892, −11.39444406785659840295054798179, −10.81432649787247119878741359402, −9.455340003639932894334431044304, −8.195568755086646375664394795785, −6.28958245470428415713827769555, −4.96519842486739728173356185343, −4.16259496289861644255572332927, 2.09328572394766256239014459189, 4.03856758209199361747145021482, 6.52066294046240890654073568815, 7.26822516705108658204771162759, 8.403014344724545275761927050696, 10.88049747824107161144551332628, 11.59984900584510176045816836111, 12.19568856407422024301210841002, 13.41101300150340336125654612711, 14.11619222559911968719802062435

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