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Properties

Label 2-1960-1960.659-c0-0-1
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: $\chi_{1960} (659, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1960,\ (\ :0),\ 0.159 + 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.035526737\)
\(L(\frac12)\) \(\approx\) \(2.035526737\)
\(L(1)\) 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

−9.144688463751660427458892502566, −8.602821381677924683820899853675, −7.75037581003572857248793973585, −6.24818833771555690850532206103, −5.84933708943762604472813537463, −5.33276096223493432531184037827, −4.19672366176243013184738271838, −3.41585618132182578684329742650, −2.31290745864426934396176790362, −1.23419652868648489835078499148, 1.96425866559032930673875826063, 2.91056186740731899467332784960, 3.96476942835908985337222054656, 4.44785950422823020544490616977, 5.81417031534702342811457878478, 6.31297837013546311658995723729, 6.94927656976346793015581407125, 7.910844460199582309224837356287, 8.386491855333306468138746815328, 9.579951376563078987206572080700

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