↑

Properties

Label 2-1960-7.4-c1-0-19
Degree $2$
Conductor $1960$
Sign $0.991 - 0.126i$
Analytic cond. $15.6506$
Root an. cond. $3.95609$
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  + (−1.28 + 2.21i)3-s + (−0.5 − 0.866i)5-s + (−1.78 − 3.08i)9-s + (1.28 − 2.21i)11-s − 5.68·13-s + 2.56·15-s + (−1.71 + 2.97i)17-s + (−0.561 − 0.972i)19-s + (2.56 + 4.43i)23-s + (−0.499 + 0.866i)25-s + 1.43·27-s + 4.56·29-s + (5.12 − 8.87i)31-s + (3.28 + 5.68i)33-s + (−4.12 − 7.14i)37-s + ⋯
L(s)  = 1  + (−0.739 + 1.28i)3-s + (−0.223 − 0.387i)5-s + (−0.593 − 1.02i)9-s + (0.386 − 0.668i)11-s − 1.57·13-s + 0.661·15-s + (−0.416 + 0.722i)17-s + (−0.128 − 0.223i)19-s + (0.534 + 0.925i)23-s + (−0.0999 + 0.173i)25-s + 0.276·27-s + 0.847·29-s + (0.920 − 1.59i)31-s + (0.571 + 0.989i)33-s + (−0.677 − 1.17i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1960\)    =    \(2^{3} \cdot 5 \cdot 7^{2}\)
Sign: $0.991 - 0.126i$
Analytic conductor: \(15.6506\)
Root analytic conductor: \(3.95609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1960} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1960,\ (\ :1/2),\ 0.991 - 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9530669229\)
\(L(\frac12)\) \(\approx\) \(0.9530669229\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 \)
good3 \( 1 + (1.28 - 2.21i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (-1.28 + 2.21i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.68T + 13T^{2} \)
17 \( 1 + (1.71 - 2.97i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.561 + 0.972i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.56 - 4.43i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.56T + 29T^{2} \)
31 \( 1 + (-5.12 + 8.87i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.12 + 7.14i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 7.12T + 41T^{2} \)
43 \( 1 - 1.12T + 43T^{2} \)
47 \( 1 + (3.28 + 5.68i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.43 + 4.22i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.56 - 13.0i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.12 + 12.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (6.12 - 10.6i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.84 - 10.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (-1.56 - 2.70i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 13.6T + 97T^{2} \)
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

−9.414158302064046135011307707329, −8.605544440577163691491832056851, −7.68874932007599125898805078286, −6.70314395505098318796910876334, −5.73039350248098011870815815232, −5.11923213507100826667271196293, −4.32853940226450346406140532239, −3.68965768926719339848888064060, −2.37110589791773745211642624128, −0.52685535124996996378738375750, 0.842665024648229601038197617345, 2.12340510956049156368630618701, 2.95009555542848352465098475576, 4.56529238409860145833987799455, 5.08260023715118558597720071206, 6.38620415015256850602101468110, 6.78384422167332843551594399696, 7.35849042824690684618771130227, 8.110980331468210803242126245949, 9.135842474055065480279778329050

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