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Properties

Label 2-960-12.11-c1-0-0
Degree $2$
Conductor $960$
Sign $-0.962 + 0.270i$
Analytic cond. $7.66563$
Root an. cond. $2.76868$
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  + (0.468 + 1.66i)3-s + i·5-s − 0.936i·7-s + (−2.56 + 1.56i)9-s − 4.27·11-s − 3.12·13-s + (−1.66 + 0.468i)15-s + 2i·17-s − 4.27i·19-s + (1.56 − 0.438i)21-s − 7.60·23-s − 25-s + (−3.80 − 3.54i)27-s + 5.12i·29-s + 2.39i·31-s + ⋯
L(s)  = 1  + (0.270 + 0.962i)3-s + 0.447i·5-s − 0.353i·7-s + (−0.853 + 0.520i)9-s − 1.28·11-s − 0.866·13-s + (−0.430 + 0.120i)15-s + 0.485i·17-s − 0.979i·19-s + (0.340 − 0.0956i)21-s − 1.58·23-s − 0.200·25-s + (−0.731 − 0.681i)27-s + 0.951i·29-s + 0.430i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.270i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.962 + 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-0.962 + 0.270i$
Analytic conductor: \(7.66563\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ -0.962 + 0.270i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0708515 - 0.514441i\)
\(L(\frac12)\) \(\approx\) \(0.0708515 - 0.514441i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.468 - 1.66i)T \)
5 \( 1 - iT \)
good7 \( 1 + 0.936iT - 7T^{2} \)
11 \( 1 + 4.27T + 11T^{2} \)
13 \( 1 + 3.12T + 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 + 4.27iT - 19T^{2} \)
23 \( 1 + 7.60T + 23T^{2} \)
29 \( 1 - 5.12iT - 29T^{2} \)
31 \( 1 - 2.39iT - 31T^{2} \)
37 \( 1 - 3.12T + 37T^{2} \)
41 \( 1 - 7.12iT - 41T^{2} \)
43 \( 1 - 1.46iT - 43T^{2} \)
47 \( 1 - 0.936T + 47T^{2} \)
53 \( 1 + 4.24iT - 53T^{2} \)
59 \( 1 + 7.19T + 59T^{2} \)
61 \( 1 - 5.12T + 61T^{2} \)
67 \( 1 - 5.20iT - 67T^{2} \)
71 \( 1 + 6.67T + 71T^{2} \)
73 \( 1 + 8.24T + 73T^{2} \)
79 \( 1 + 9.06iT - 79T^{2} \)
83 \( 1 + 4.68T + 83T^{2} \)
89 \( 1 + 6.24iT - 89T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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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

−10.31451560930456117003751597592, −9.917005890248448730870882011222, −8.876911770431693587711451123097, −7.987033092357142138776598973052, −7.29604990511367715162464031316, −6.06706060957325149524766765799, −5.09749717855156426175664494501, −4.33219679901458911859269677023, −3.17699623539425017319657930453, −2.33277974300831432861263484361, 0.21040371494137821467693618528, 1.97298467338600515819011028300, 2.75319910985165030082078448235, 4.18890289571982489832558064048, 5.47415512364371625223206439272, 5.98490463393112318434300932258, 7.30567722344276053131550097374, 7.85583284513304335230876608434, 8.512076166079678277034738567214, 9.558809028366651136232689852463

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