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Properties

Label 2-960-12.11-c1-0-1
Degree $2$
Conductor $960$
Sign $-0.707 - 0.707i$
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  + (−1.22 + 1.22i)3-s − i·5-s − 2.44i·7-s − 2.99i·9-s − 4.89·11-s + 2·13-s + (1.22 + 1.22i)15-s + 6i·17-s + 4.89i·19-s + (2.99 + 2.99i)21-s − 2.44·23-s − 25-s + (3.67 + 3.67i)27-s + 9.79i·31-s + (5.99 − 5.99i)33-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)3-s − 0.447i·5-s − 0.925i·7-s − 0.999i·9-s − 1.47·11-s + 0.554·13-s + (0.316 + 0.316i)15-s + 1.45i·17-s + 1.12i·19-s + (0.654 + 0.654i)21-s − 0.510·23-s − 0.200·25-s + (0.707 + 0.707i)27-s + 1.75i·31-s + (1.04 − 1.04i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-0.707 - 0.707i$
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: \((2,\ 960,\ (\ :1/2),\ -0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.189481 + 0.457448i\)
\(L(\frac12)\) \(\approx\) \(0.189481 + 0.457448i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.22 - 1.22i)T \)
5 \( 1 + iT \)
good7 \( 1 + 2.44iT - 7T^{2} \)
11 \( 1 + 4.89T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 - 6iT - 17T^{2} \)
19 \( 1 - 4.89iT - 19T^{2} \)
23 \( 1 + 2.44T + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 9.79iT - 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 - 6iT - 41T^{2} \)
43 \( 1 + 2.44iT - 43T^{2} \)
47 \( 1 + 12.2T + 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 - 9.79T + 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 - 7.34iT - 67T^{2} \)
71 \( 1 - 4.89T + 71T^{2} \)
73 \( 1 + 14T + 73T^{2} \)
79 \( 1 - 4.89iT - 79T^{2} \)
83 \( 1 + 7.34T + 83T^{2} \)
89 \( 1 - 12iT - 89T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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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

−10.36980428763196612538326603501, −9.894786557293250740948932817730, −8.544596059939490590832656158471, −8.024950369798289622693053655832, −6.81662078535189697027134832886, −5.87572119347522519925708528914, −5.12472752068638653639590772684, −4.15962992573768230986221690147, −3.37163859595033019658645270915, −1.43114034423143588713307255564, 0.25335799541804009280987218542, 2.19313063980522454411793367051, 2.90276321336599423576367816579, 4.70888123122553129579711285619, 5.49143818068792493144645817077, 6.19413049295776319639520865243, 7.22064748074488154411461570814, 7.82972994384251404404456904052, 8.803201303797590131797384890815, 9.816419980798828182272515923052

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