↑

Properties

Label 2-65-65.32-c1-0-4
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

Related objects

Downloads

Learn more

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: $\chi_{65} (32, \cdot )$
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 \( 1 + (3.75 + 6.50i)T + (-48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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