↑

Properties

Label 2-1960-7.2-c1-0-23
Degree $2$
Conductor $1960$
Sign $0.827 + 0.561i$
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  + (0.207 + 0.358i)3-s + (0.5 − 0.866i)5-s + (1.41 − 2.44i)9-s + (0.5 + 0.866i)11-s + 2.41·13-s + 0.414·15-s + (0.207 + 0.358i)17-s + (−1 + 1.73i)19-s + (1.12 − 1.94i)23-s + (−0.499 − 0.866i)25-s + 2.41·27-s + 29-s + (−0.878 − 1.52i)31-s + (−0.207 + 0.358i)33-s + (3.94 − 6.84i)37-s + ⋯
L(s)  = 1  + (0.119 + 0.207i)3-s + (0.223 − 0.387i)5-s + (0.471 − 0.816i)9-s + (0.150 + 0.261i)11-s + 0.669·13-s + 0.106·15-s + (0.0502 + 0.0870i)17-s + (−0.229 + 0.397i)19-s + (0.233 − 0.404i)23-s + (−0.0999 − 0.173i)25-s + 0.464·27-s + 0.185·29-s + (−0.157 − 0.273i)31-s + (−0.0360 + 0.0624i)33-s + (0.649 − 1.12i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1960\)    =    \(2^{3} \cdot 5 \cdot 7^{2}\)
Sign: $0.827 + 0.561i$
Analytic conductor: \(15.6506\)
Root analytic conductor: \(3.95609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1960} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1960,\ (\ :1/2),\ 0.827 + 0.561i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.044492425\)
\(L(\frac12)\) \(\approx\) \(2.044492425\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.207 - 0.358i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.41T + 13T^{2} \)
17 \( 1 + (-0.207 - 0.358i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.12 + 1.94i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - T + 29T^{2} \)
31 \( 1 + (0.878 + 1.52i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.94 + 6.84i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 7.41T + 41T^{2} \)
43 \( 1 + 0.343T + 43T^{2} \)
47 \( 1 + (5.20 - 9.01i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.70 + 8.15i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.12 + 8.87i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.585 + 1.01i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.707 + 1.22i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 14.4T + 71T^{2} \)
73 \( 1 + (-2.58 - 4.47i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.32 + 12.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 11.3T + 83T^{2} \)
89 \( 1 + (-0.828 + 1.43i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 0.0710T + 97T^{2} \)
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

−9.374472963439937907887806968645, −8.366327006057139316147703151384, −7.65557974866650382983709090101, −6.55271233482887311668818034616, −6.08545876644797208834402016467, −4.96086590701294201346008071561, −4.13456675316856156512176725016, −3.36813422973163646227428615269, −2.03795226794678547387537672081, −0.855584788905528014382787503215, 1.21247644087052288830120455479, 2.32843151174142200171354169124, 3.29811471289852055506978200965, 4.35771679794329488623745201294, 5.24285683449288367217018594545, 6.19079110709963426876900338950, 6.89002536876913464329688082957, 7.71463086012133875810785292571, 8.391785224965705071364148110945, 9.258870768094333285637780114035

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