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Properties

Label 2-960-960.269-c0-0-1
Degree $2$
Conductor $960$
Sign $0.956 - 0.290i$
Analytic cond. $0.479102$
Root an. cond. $0.692172$
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  + (−0.195 + 0.980i)2-s + (0.831 − 0.555i)3-s + (−0.923 − 0.382i)4-s + (0.980 − 0.195i)5-s + (0.382 + 0.923i)6-s + (0.555 − 0.831i)8-s + (0.382 − 0.923i)9-s + i·10-s + (−0.980 + 0.195i)12-s + (0.707 − 0.707i)15-s + (0.707 + 0.707i)16-s + (−1.38 − 1.38i)17-s + (0.831 + 0.555i)18-s + (−0.216 + 1.08i)19-s + (−0.980 − 0.195i)20-s + ⋯
L(s)  = 1  + (−0.195 + 0.980i)2-s + (0.831 − 0.555i)3-s + (−0.923 − 0.382i)4-s + (0.980 − 0.195i)5-s + (0.382 + 0.923i)6-s + (0.555 − 0.831i)8-s + (0.382 − 0.923i)9-s + i·10-s + (−0.980 + 0.195i)12-s + (0.707 − 0.707i)15-s + (0.707 + 0.707i)16-s + (−1.38 − 1.38i)17-s + (0.831 + 0.555i)18-s + (−0.216 + 1.08i)19-s + (−0.980 − 0.195i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.956 - 0.290i$
Analytic conductor: \(0.479102\)
Root analytic conductor: \(0.692172\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :0),\ 0.956 - 0.290i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.248578380\)
\(L(\frac12)\) \(\approx\) \(1.248578380\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.195 - 0.980i)T \)
3 \( 1 + (-0.831 + 0.555i)T \)
5 \( 1 + (-0.980 + 0.195i)T \)
good7 \( 1 + (0.707 - 0.707i)T^{2} \)
11 \( 1 + (0.382 + 0.923i)T^{2} \)
13 \( 1 + (-0.923 - 0.382i)T^{2} \)
17 \( 1 + (1.38 + 1.38i)T + iT^{2} \)
19 \( 1 + (0.216 - 1.08i)T + (-0.923 - 0.382i)T^{2} \)
23 \( 1 + (0.360 + 0.149i)T + (0.707 + 0.707i)T^{2} \)
29 \( 1 + (0.382 - 0.923i)T^{2} \)
31 \( 1 - 1.84iT - T^{2} \)
37 \( 1 + (0.923 - 0.382i)T^{2} \)
41 \( 1 + (-0.707 - 0.707i)T^{2} \)
43 \( 1 + (-0.382 - 0.923i)T^{2} \)
47 \( 1 + (-1.17 - 1.17i)T + iT^{2} \)
53 \( 1 + (0.425 - 0.636i)T + (-0.382 - 0.923i)T^{2} \)
59 \( 1 + (-0.923 + 0.382i)T^{2} \)
61 \( 1 + (-0.923 + 0.617i)T + (0.382 - 0.923i)T^{2} \)
67 \( 1 + (-0.382 + 0.923i)T^{2} \)
71 \( 1 + (0.707 - 0.707i)T^{2} \)
73 \( 1 + (0.707 + 0.707i)T^{2} \)
79 \( 1 + (1 - i)T - iT^{2} \)
83 \( 1 + (-0.149 + 0.750i)T + (-0.923 - 0.382i)T^{2} \)
89 \( 1 + (-0.707 + 0.707i)T^{2} \)
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

−9.853707363287675848902197133692, −9.178514027191747004828402249546, −8.652858141990657970756101795716, −7.74430618963198388583800835426, −6.83787603842275751557321520410, −6.28576833617730507630132936869, −5.21592302356782620196135050640, −4.21354241241027332760214610515, −2.75969476923195816037271619409, −1.45832731178677814512245356744, 1.93930814312629708338747869783, 2.52641905046569448820966043719, 3.78605830574531027549749174681, 4.56082797500166697892004162595, 5.65857110441289903918475932802, 6.89042203985823997692921466181, 8.135381427092924445131068106835, 8.812113211353761383045791250777, 9.444196495218162976854878256473, 10.16203082594990455231089736433

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