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Properties

Label 2-1960-7.4-c1-0-35
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.20 − 2.09i)3-s + (−0.5 − 0.866i)5-s + (−1.41 − 2.44i)9-s + (0.5 − 0.866i)11-s + 0.414·13-s − 2.41·15-s + (1.20 − 2.09i)17-s + (1 + 1.73i)19-s + (−3.12 − 5.40i)23-s + (−0.499 + 0.866i)25-s + 0.414·27-s + 29-s + (5.12 − 8.87i)31-s + (−1.20 − 2.09i)33-s + (−5.94 − 10.3i)37-s + ⋯
L(s)  = 1  + (0.696 − 1.20i)3-s + (−0.223 − 0.387i)5-s + (−0.471 − 0.816i)9-s + (0.150 − 0.261i)11-s + 0.114·13-s − 0.623·15-s + (0.292 − 0.507i)17-s + (0.229 + 0.397i)19-s + (−0.650 − 1.12i)23-s + (−0.0999 + 0.173i)25-s + 0.0797·27-s + 0.185·29-s + (0.919 − 1.59i)31-s + (−0.210 − 0.363i)33-s + (−0.978 − 1.69i)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: $\chi_{1960} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1960,\ (\ :1/2),\ -0.827 + 0.561i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.854562810\)
\(L(\frac12)\) \(\approx\) \(1.854562810\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.20 + 2.09i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 0.414T + 13T^{2} \)
17 \( 1 + (-1.20 + 2.09i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.12 + 5.40i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - T + 29T^{2} \)
31 \( 1 + (-5.12 + 8.87i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.94 + 10.3i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.58T + 41T^{2} \)
43 \( 1 + 11.6T + 43T^{2} \)
47 \( 1 + (-3.79 - 6.56i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.29 - 5.70i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.878 + 1.52i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.41 + 5.91i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.707 + 1.22i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.48T + 71T^{2} \)
73 \( 1 + (5.41 - 9.37i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.67 - 2.89i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 11.3T + 83T^{2} \)
89 \( 1 + (-4.82 - 8.36i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 14.0T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.601419168271006541989618583227, −8.075140889681761120572188371690, −7.45714289506006934210304127108, −6.61911396416858915494472260593, −5.87687592712878200582556584380, −4.75909047303676119279642096932, −3.72241990104102566286019176269, −2.69322064924107895466724565350, −1.77289942591159686217517774889, −0.60779134447075641132921764215, 1.68406612118923734709058021469, 3.17169697004422722589404068773, 3.46664150563461534066233679401, 4.55937521337716759673232374726, 5.19546786715997180017672272889, 6.40916477521136860935782732972, 7.16906198801043187002274497987, 8.272390042749553860310656613261, 8.647047823804666871473109528661, 9.631824866108135816151479592519

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