Properties

Label 2-1960-7.4-c1-0-25
Degree $2$
Conductor $1960$
Sign $0.605 + 0.795i$
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  + (1.29 − 2.23i)3-s + (0.5 + 0.866i)5-s + (−1.83 − 3.18i)9-s + (−0.839 + 1.45i)11-s + 4.84·13-s + 2.58·15-s + (1 − 1.73i)17-s + (3.42 + 5.92i)19-s + (1.13 + 1.95i)23-s + (−0.499 + 0.866i)25-s − 1.75·27-s + 3.32·29-s + (4.58 − 7.94i)31-s + (2.16 + 3.75i)33-s + (1.42 + 2.46i)37-s + ⋯
L(s)  = 1  + (0.746 − 1.29i)3-s + (0.223 + 0.387i)5-s + (−0.613 − 1.06i)9-s + (−0.253 + 0.438i)11-s + 1.34·13-s + 0.667·15-s + (0.242 − 0.420i)17-s + (0.785 + 1.36i)19-s + (0.235 + 0.408i)23-s + (−0.0999 + 0.173i)25-s − 0.337·27-s + 0.616·29-s + (0.823 − 1.42i)31-s + (0.377 + 0.653i)33-s + (0.233 + 0.405i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1960\)    =    \(2^{3} \cdot 5 \cdot 7^{2}\)
Sign: $0.605 + 0.795i$
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: \((2,\ 1960,\ (\ :1/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.606920492\)
\(L(\frac12)\) \(\approx\) \(2.606920492\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.29 + 2.23i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (0.839 - 1.45i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.84T + 13T^{2} \)
17 \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.42 - 5.92i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.13 - 1.95i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.32T + 29T^{2} \)
31 \( 1 + (-4.58 + 7.94i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.42 - 2.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 9.52T + 41T^{2} \)
43 \( 1 - 6.58T + 43T^{2} \)
47 \( 1 + (6.10 + 10.5i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.74 - 6.48i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4 + 6.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.24 + 5.62i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.87 + 4.98i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (5.84 - 10.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.84 + 4.93i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 12.5T + 83T^{2} \)
89 \( 1 + (2.92 + 5.06i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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

−8.887212308872020648565565107102, −7.979787523652368687284543803565, −7.74723344792843669507407570720, −6.70296564184533675174050377282, −6.20126455560215396979961995596, −5.19934938951959715654996334085, −3.77954865524536929482978870565, −3.00127615710376660319121463950, −1.98450117000370584814677719430, −1.11354062419545433122464551097, 1.16963977662724370370158520726, 2.81406393441930216580889252527, 3.38135196367037019280944439475, 4.40359747244801973447943750512, 5.02321695872046806588165510405, 5.96904161205220414073053815316, 6.91737512514939515784365315121, 8.208418567259846324735784569332, 8.608148263433659798544342935840, 9.215706138470232805701654789428

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