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Properties

Label 2-1960-1960.1499-c0-0-0
Degree $2$
Conductor $1960$
Sign $0.159 - 0.987i$
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  + (−0.900 − 0.433i)2-s + (0.623 + 0.781i)4-s + (−0.222 − 0.974i)5-s + (−0.222 + 0.974i)7-s + (−0.222 − 0.974i)8-s + (−0.900 + 0.433i)9-s + (−0.222 + 0.974i)10-s + (0.400 + 0.193i)11-s + (−1.12 − 0.541i)13-s + (0.623 − 0.781i)14-s + (−0.222 + 0.974i)16-s + 18-s − 0.445·19-s + (0.623 − 0.781i)20-s + (−0.277 − 0.347i)22-s + (1.24 + 1.56i)23-s + ⋯
L(s)  = 1  + (−0.900 − 0.433i)2-s + (0.623 + 0.781i)4-s + (−0.222 − 0.974i)5-s + (−0.222 + 0.974i)7-s + (−0.222 − 0.974i)8-s + (−0.900 + 0.433i)9-s + (−0.222 + 0.974i)10-s + (0.400 + 0.193i)11-s + (−1.12 − 0.541i)13-s + (0.623 − 0.781i)14-s + (−0.222 + 0.974i)16-s + 18-s − 0.445·19-s + (0.623 − 0.781i)20-s + (−0.277 − 0.347i)22-s + (1.24 + 1.56i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.159 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.159 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1960\)    =    \(2^{3} \cdot 5 \cdot 7^{2}\)
Sign: $0.159 - 0.987i$
Analytic conductor: \(0.978167\)
Root analytic conductor: \(0.989023\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1960} (1499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1960,\ (\ :0),\ 0.159 - 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3828416159\)
\(L(\frac12)\) \(\approx\) \(0.3828416159\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.900 + 0.433i)T \)
5 \( 1 + (0.222 + 0.974i)T \)
7 \( 1 + (0.222 - 0.974i)T \)
good3 \( 1 + (0.900 - 0.433i)T^{2} \)
11 \( 1 + (-0.400 - 0.193i)T + (0.623 + 0.781i)T^{2} \)
13 \( 1 + (1.12 + 0.541i)T + (0.623 + 0.781i)T^{2} \)
17 \( 1 + (0.222 + 0.974i)T^{2} \)
19 \( 1 + 0.445T + T^{2} \)
23 \( 1 + (-1.24 - 1.56i)T + (-0.222 + 0.974i)T^{2} \)
29 \( 1 + (0.222 + 0.974i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (1.12 - 1.40i)T + (-0.222 - 0.974i)T^{2} \)
41 \( 1 + (-0.400 - 1.75i)T + (-0.900 + 0.433i)T^{2} \)
43 \( 1 + (0.900 + 0.433i)T^{2} \)
47 \( 1 + (1.12 + 0.541i)T + (0.623 + 0.781i)T^{2} \)
53 \( 1 + (-0.777 - 0.974i)T + (-0.222 + 0.974i)T^{2} \)
59 \( 1 + (-0.0990 + 0.433i)T + (-0.900 - 0.433i)T^{2} \)
61 \( 1 + (0.222 + 0.974i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.222 - 0.974i)T^{2} \)
73 \( 1 + (-0.623 + 0.781i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (-0.623 + 0.781i)T^{2} \)
89 \( 1 + (1.12 - 0.541i)T + (0.623 - 0.781i)T^{2} \)
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

−9.447082411626092191674807201329, −8.812103242554722437079742963617, −8.219562162004080108734478756484, −7.53562690995676497221189472512, −6.50665828882715130792891636765, −5.44916143953632755035180677129, −4.81693924782931973767659098310, −3.38862365959754102096332096062, −2.60298448036365399077808041365, −1.46886413022833740938864302673, 0.36419464783361590699375675505, 2.18736291579561490883190568197, 3.11424526253077829867377435001, 4.22861269407791217066024936612, 5.41763613511590976108063111829, 6.46880797866279704928011922014, 6.90884078724578730179065407363, 7.44974754045405882087731278015, 8.496764455028608081947012150770, 9.103665212627742174155275200381

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