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Properties

Label 2-1960-1960.779-c0-0-1
Degree $2$
Conductor $1960$
Sign $0.910 - 0.414i$
Analytic cond. $0.978167$
Root an. cond. $0.989023$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.988 − 0.149i)2-s + (0.955 − 0.294i)4-s + (−0.0747 + 0.997i)5-s + (0.900 + 0.433i)7-s + (0.900 − 0.433i)8-s + (0.365 + 0.930i)9-s + (0.0747 + 0.997i)10-s + (0.0546 − 0.139i)11-s + (−1.19 − 1.49i)13-s + (0.955 + 0.294i)14-s + (0.826 − 0.563i)16-s + (0.5 + 0.866i)18-s + (−0.826 + 1.43i)19-s + (0.222 + 0.974i)20-s + (0.0332 − 0.145i)22-s + (−0.733 − 0.680i)23-s + ⋯
L(s)  = 1  + (0.988 − 0.149i)2-s + (0.955 − 0.294i)4-s + (−0.0747 + 0.997i)5-s + (0.900 + 0.433i)7-s + (0.900 − 0.433i)8-s + (0.365 + 0.930i)9-s + (0.0747 + 0.997i)10-s + (0.0546 − 0.139i)11-s + (−1.19 − 1.49i)13-s + (0.955 + 0.294i)14-s + (0.826 − 0.563i)16-s + (0.5 + 0.866i)18-s + (−0.826 + 1.43i)19-s + (0.222 + 0.974i)20-s + (0.0332 − 0.145i)22-s + (−0.733 − 0.680i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1960\)    =    \(2^{3} \cdot 5 \cdot 7^{2}\)
Sign: $0.910 - 0.414i$
Analytic conductor: \(0.978167\)
Root analytic conductor: \(0.989023\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1960} (779, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1960,\ (\ :0),\ 0.910 - 0.414i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.273211829\)
\(L(\frac12)\) \(\approx\) \(2.273211829\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.988 + 0.149i)T \)
5 \( 1 + (0.0747 - 0.997i)T \)
7 \( 1 + (-0.900 - 0.433i)T \)
good3 \( 1 + (-0.365 - 0.930i)T^{2} \)
11 \( 1 + (-0.0546 + 0.139i)T + (-0.733 - 0.680i)T^{2} \)
13 \( 1 + (1.19 + 1.49i)T + (-0.222 + 0.974i)T^{2} \)
17 \( 1 + (-0.0747 + 0.997i)T^{2} \)
19 \( 1 + (0.826 - 1.43i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.733 + 0.680i)T + (0.0747 + 0.997i)T^{2} \)
29 \( 1 + (0.900 - 0.433i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.698 + 0.215i)T + (0.826 + 0.563i)T^{2} \)
41 \( 1 + (-1.78 + 0.858i)T + (0.623 - 0.781i)T^{2} \)
43 \( 1 + (-0.623 - 0.781i)T^{2} \)
47 \( 1 + (1.44 - 0.218i)T + (0.955 - 0.294i)T^{2} \)
53 \( 1 + (-1.40 + 0.432i)T + (0.826 - 0.563i)T^{2} \)
59 \( 1 + (0.134 + 1.79i)T + (-0.988 + 0.149i)T^{2} \)
61 \( 1 + (-0.826 - 0.563i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.900 + 0.433i)T^{2} \)
73 \( 1 + (-0.955 - 0.294i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.222 + 0.974i)T^{2} \)
89 \( 1 + (0.162 + 0.414i)T + (-0.733 + 0.680i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.850412976418588435274444868186, −8.141100748779951269798407206734, −7.85904109530871676322306215478, −7.05347764446757269453862018642, −6.00034504616383730681357545042, −5.40599427689413944112967990664, −4.57142864884132699809158375813, −3.63082578583841859265758325240, −2.50779091896232345230157991087, −1.98286172914251866891171905293, 1.43005566045949170484536652967, 2.40039448696630142766562895486, 3.96473650719910331373391963847, 4.44228666518919375052905689917, 4.98194230447370873082511917369, 6.06579752389646027272167959906, 6.99918029881634976666288830890, 7.45871206141260147636328481954, 8.505201924143412278547595446734, 9.254387310318256954481229406881

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