↑

Properties

Label 2-639-71.20-c1-0-18
Degree $2$
Conductor $639$
Sign $0.559 + 0.828i$
Analytic cond. $5.10244$
Root an. cond. $2.25885$
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  + (0.0812 + 0.356i)2-s + (1.68 − 0.809i)4-s − 3.28·5-s + (−0.0815 + 0.357i)7-s + (0.880 + 1.10i)8-s + (−0.266 − 1.16i)10-s + (2.34 − 2.94i)11-s + (−3.55 − 4.45i)13-s − 0.133·14-s + (2.00 − 2.51i)16-s + 3.51·17-s + (3.05 + 1.47i)19-s + (−5.52 + 2.65i)20-s + (1.23 + 0.596i)22-s + (1.34 − 5.91i)23-s + ⋯
L(s)  = 1  + (0.0574 + 0.251i)2-s + (0.840 − 0.404i)4-s − 1.46·5-s + (−0.0308 + 0.135i)7-s + (0.311 + 0.390i)8-s + (−0.0844 − 0.369i)10-s + (0.707 − 0.887i)11-s + (−0.986 − 1.23i)13-s − 0.0357·14-s + (0.501 − 0.628i)16-s + 0.852·17-s + (0.701 + 0.338i)19-s + (−1.23 + 0.594i)20-s + (0.264 + 0.127i)22-s + (0.281 − 1.23i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.559 + 0.828i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.559 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(639\)    =    \(3^{2} \cdot 71\)
Sign: $0.559 + 0.828i$
Analytic conductor: \(5.10244\)
Root analytic conductor: \(2.25885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{639} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 639,\ (\ :1/2),\ 0.559 + 0.828i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21709 - 0.646742i\)
\(L(\frac12)\) \(\approx\) \(1.21709 - 0.646742i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
71 \( 1 + (-7.05 + 4.60i)T \)
good2 \( 1 + (-0.0812 - 0.356i)T + (-1.80 + 0.867i)T^{2} \)
5 \( 1 + 3.28T + 5T^{2} \)
7 \( 1 + (0.0815 - 0.357i)T + (-6.30 - 3.03i)T^{2} \)
11 \( 1 + (-2.34 + 2.94i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (3.55 + 4.45i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 - 3.51T + 17T^{2} \)
19 \( 1 + (-3.05 - 1.47i)T + (11.8 + 14.8i)T^{2} \)
23 \( 1 + (-1.34 + 5.91i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (-2.00 + 0.963i)T + (18.0 - 22.6i)T^{2} \)
31 \( 1 + (4.90 + 6.14i)T + (-6.89 + 30.2i)T^{2} \)
37 \( 1 + (-1.39 - 6.11i)T + (-33.3 + 16.0i)T^{2} \)
41 \( 1 + (0.959 + 1.20i)T + (-9.12 + 39.9i)T^{2} \)
43 \( 1 + (1.65 + 7.25i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (-1.13 - 0.548i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (10.4 + 5.03i)T + (33.0 + 41.4i)T^{2} \)
59 \( 1 + (5.26 - 6.60i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (-0.0921 - 0.403i)T + (-54.9 + 26.4i)T^{2} \)
67 \( 1 + (-3.71 + 1.78i)T + (41.7 - 52.3i)T^{2} \)
73 \( 1 + (-2.77 - 12.1i)T + (-65.7 + 31.6i)T^{2} \)
79 \( 1 + (-1.56 - 1.95i)T + (-17.5 + 77.0i)T^{2} \)
83 \( 1 + (7.45 - 9.35i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (-5.99 - 2.88i)T + (55.4 + 69.5i)T^{2} \)
97 \( 1 + (3.55 - 4.46i)T + (-21.5 - 94.5i)T^{2} \)
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

−10.61092752048069101341298836914, −9.694104763920100773871772527047, −8.330121354218632156147055268041, −7.78419165624327907078783587887, −7.02243920126103508542615726910, −5.94486569889189661608947901892, −5.04023835582023159143241168894, −3.68684146150874495281555227539, −2.77057170618736738230887511262, −0.77211221573226440772844881688, 1.62139630505034667900667362549, 3.15738831788351155423255187310, 3.96275740857844933746052193457, 4.93193123440888608401602987653, 6.60344630802478448262297025371, 7.43991436426368820082791417466, 7.60254770873306898632903696873, 9.046485516388865961916388041335, 9.862055305017463540970741451971, 11.05621629835076085798432307200

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