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Properties

Label 2-1960-1960.739-c0-0-1
Degree $2$
Conductor $1960$
Sign $0.775 - 0.631i$
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  + (0.826 + 0.563i)2-s + (0.365 + 0.930i)4-s + (0.955 − 0.294i)5-s + (−0.222 − 0.974i)7-s + (−0.222 + 0.974i)8-s + (0.0747 + 0.997i)9-s + (0.955 + 0.294i)10-s + (0.142 − 1.90i)11-s + (−0.658 + 0.317i)13-s + (0.365 − 0.930i)14-s + (−0.733 + 0.680i)16-s + (−0.5 + 0.866i)18-s + (0.733 + 1.26i)19-s + (0.623 + 0.781i)20-s + (1.19 − 1.49i)22-s + (0.988 + 0.149i)23-s + ⋯
L(s)  = 1  + (0.826 + 0.563i)2-s + (0.365 + 0.930i)4-s + (0.955 − 0.294i)5-s + (−0.222 − 0.974i)7-s + (−0.222 + 0.974i)8-s + (0.0747 + 0.997i)9-s + (0.955 + 0.294i)10-s + (0.142 − 1.90i)11-s + (−0.658 + 0.317i)13-s + (0.365 − 0.930i)14-s + (−0.733 + 0.680i)16-s + (−0.5 + 0.866i)18-s + (0.733 + 1.26i)19-s + (0.623 + 0.781i)20-s + (1.19 − 1.49i)22-s + (0.988 + 0.149i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1960\)    =    \(2^{3} \cdot 5 \cdot 7^{2}\)
Sign: $0.775 - 0.631i$
Analytic conductor: \(0.978167\)
Root analytic conductor: \(0.989023\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1960} (739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1960,\ (\ :0),\ 0.775 - 0.631i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.103137439\)
\(L(\frac12)\) \(\approx\) \(2.103137439\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.826 - 0.563i)T \)
5 \( 1 + (-0.955 + 0.294i)T \)
7 \( 1 + (0.222 + 0.974i)T \)
good3 \( 1 + (-0.0747 - 0.997i)T^{2} \)
11 \( 1 + (-0.142 + 1.90i)T + (-0.988 - 0.149i)T^{2} \)
13 \( 1 + (0.658 - 0.317i)T + (0.623 - 0.781i)T^{2} \)
17 \( 1 + (-0.955 + 0.294i)T^{2} \)
19 \( 1 + (-0.733 - 1.26i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.988 - 0.149i)T + (0.955 + 0.294i)T^{2} \)
29 \( 1 + (0.222 - 0.974i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.0546 + 0.139i)T + (-0.733 - 0.680i)T^{2} \)
41 \( 1 + (0.367 - 1.61i)T + (-0.900 - 0.433i)T^{2} \)
43 \( 1 + (0.900 - 0.433i)T^{2} \)
47 \( 1 + (1.63 + 1.11i)T + (0.365 + 0.930i)T^{2} \)
53 \( 1 + (0.722 + 1.84i)T + (-0.733 + 0.680i)T^{2} \)
59 \( 1 + (0.425 + 0.131i)T + (0.826 + 0.563i)T^{2} \)
61 \( 1 + (0.733 + 0.680i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.222 + 0.974i)T^{2} \)
73 \( 1 + (-0.365 + 0.930i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.623 - 0.781i)T^{2} \)
89 \( 1 + (-0.0931 - 1.24i)T + (-0.988 + 0.149i)T^{2} \)
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

−9.450090909085317583502923969622, −8.339982407053819405713765608903, −7.930402733862081069448368589650, −6.87289745599136665121481407744, −6.25176360005512652736210300259, −5.35419720004468108150873843714, −4.86976608839060209052735413669, −3.65453754812580415529977738248, −2.92009361612919017854669744224, −1.54898356741423585301907957369, 1.55117653899820441067323096229, 2.55463560598446481970437500027, 3.12996077614475514112400756830, 4.57523340924544049239509478237, 5.11654767672703109700077183846, 5.99015249768247762719521933898, 6.81449196341461663440998471515, 7.25355204688217596625218469411, 9.053909480299334919186079189404, 9.462329892805855392478606093171

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