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Properties

Label 2-960-192.131-c1-0-92
Degree $2$
Conductor $960$
Sign $0.518 + 0.855i$
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  + (1.39 − 0.242i)2-s + (−1.69 + 0.344i)3-s + (1.88 − 0.674i)4-s + (−0.555 + 0.831i)5-s + (−2.28 + 0.891i)6-s + (−0.799 − 1.93i)7-s + (2.46 − 1.39i)8-s + (2.76 − 1.17i)9-s + (−0.572 + 1.29i)10-s + (−0.887 + 0.176i)11-s + (−2.96 + 1.79i)12-s + (−1.65 + 1.10i)13-s + (−1.58 − 2.49i)14-s + (0.656 − 1.60i)15-s + (3.09 − 2.53i)16-s + (5.29 − 5.29i)17-s + ⋯
L(s)  = 1  + (0.985 − 0.171i)2-s + (−0.979 + 0.199i)3-s + (0.941 − 0.337i)4-s + (−0.248 + 0.371i)5-s + (−0.931 + 0.363i)6-s + (−0.302 − 0.729i)7-s + (0.869 − 0.493i)8-s + (0.920 − 0.390i)9-s + (−0.181 + 0.408i)10-s + (−0.267 + 0.0532i)11-s + (−0.855 + 0.517i)12-s + (−0.458 + 0.306i)13-s + (−0.422 − 0.667i)14-s + (0.169 − 0.413i)15-s + (0.772 − 0.634i)16-s + (1.28 − 1.28i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.518 + 0.855i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.518 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.518 + 0.855i$
Analytic conductor: \(7.66563\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ 0.518 + 0.855i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73327 - 0.976180i\)
\(L(\frac12)\) \(\approx\) \(1.73327 - 0.976180i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 + 0.242i)T \)
3 \( 1 + (1.69 - 0.344i)T \)
5 \( 1 + (0.555 - 0.831i)T \)
good7 \( 1 + (0.799 + 1.93i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (0.887 - 0.176i)T + (10.1 - 4.20i)T^{2} \)
13 \( 1 + (1.65 - 1.10i)T + (4.97 - 12.0i)T^{2} \)
17 \( 1 + (-5.29 + 5.29i)T - 17iT^{2} \)
19 \( 1 + (-0.695 + 0.464i)T + (7.27 - 17.5i)T^{2} \)
23 \( 1 + (-0.779 + 1.88i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (-1.12 + 5.66i)T + (-26.7 - 11.0i)T^{2} \)
31 \( 1 + 0.323T + 31T^{2} \)
37 \( 1 + (-1.85 + 2.77i)T + (-14.1 - 34.1i)T^{2} \)
41 \( 1 + (-8.87 - 3.67i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (-2.67 + 0.532i)T + (39.7 - 16.4i)T^{2} \)
47 \( 1 + (5.05 + 5.05i)T + 47iT^{2} \)
53 \( 1 + (0.647 + 3.25i)T + (-48.9 + 20.2i)T^{2} \)
59 \( 1 + (1.86 + 1.24i)T + (22.5 + 54.5i)T^{2} \)
61 \( 1 + (2.77 - 13.9i)T + (-56.3 - 23.3i)T^{2} \)
67 \( 1 + (-0.225 - 0.0449i)T + (61.8 + 25.6i)T^{2} \)
71 \( 1 + (-1.58 + 0.655i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (-1.69 - 0.701i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (-2.86 - 2.86i)T + 79iT^{2} \)
83 \( 1 + (5.69 + 8.52i)T + (-31.7 + 76.6i)T^{2} \)
89 \( 1 + (-0.568 + 0.235i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 - 7.81iT - 97T^{2} \)
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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

−10.11408058999559845894429191991, −9.579668740398040314768241286012, −7.64902129959884283616968826284, −7.19489699414337868432883768516, −6.33026314187715200776645868013, −5.43528840492025233345821144114, −4.60142917662427137464623649772, −3.77109693774041203599678420367, −2.63659102289860724111370362951, −0.821422592071049852245784866133, 1.47795922054609001343153478405, 2.95880787173423465379791919440, 4.09960121213531296193956204911, 5.16924660334762446476338441285, 5.68528088213160774424756579842, 6.43516522734309594703572895204, 7.52470427589987912964760014510, 8.109651056478738777971748081770, 9.482426788343091424294424294331, 10.47955270567137602929662342944

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