Properties

Label 2-960-24.11-c1-0-1
Degree $2$
Conductor $960$
Sign $-0.580 - 0.814i$
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  + (−1.10 + 1.33i)3-s − 5-s − 4.67i·7-s + (−0.571 − 2.94i)9-s + 1.14i·11-s + 2.42i·13-s + (1.10 − 1.33i)15-s + 7.87i·17-s − 2.42·19-s + (6.24 + 5.14i)21-s − 4.62·23-s + 25-s + (4.56 + 2.48i)27-s + 6.48·29-s + 3.14i·31-s + ⋯
L(s)  = 1  + (−0.636 + 0.771i)3-s − 0.447·5-s − 1.76i·7-s + (−0.190 − 0.981i)9-s + 0.344i·11-s + 0.672i·13-s + (0.284 − 0.345i)15-s + 1.90i·17-s − 0.556·19-s + (1.36 + 1.12i)21-s − 0.965·23-s + 0.200·25-s + (0.878 + 0.477i)27-s + 1.20·29-s + 0.564i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.580 - 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.580 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-0.580 - 0.814i$
Analytic conductor: \(7.66563\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (671, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ -0.580 - 0.814i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.280710 + 0.544984i\)
\(L(\frac12)\) \(\approx\) \(0.280710 + 0.544984i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.10 - 1.33i)T \)
5 \( 1 + T \)
good7 \( 1 + 4.67iT - 7T^{2} \)
11 \( 1 - 1.14iT - 11T^{2} \)
13 \( 1 - 2.42iT - 13T^{2} \)
17 \( 1 - 7.87iT - 17T^{2} \)
19 \( 1 + 2.42T + 19T^{2} \)
23 \( 1 + 4.62T + 23T^{2} \)
29 \( 1 - 6.48T + 29T^{2} \)
31 \( 1 - 3.14iT - 31T^{2} \)
37 \( 1 - 6.83iT - 37T^{2} \)
41 \( 1 - 5.88iT - 41T^{2} \)
43 \( 1 + 6.61T + 43T^{2} \)
47 \( 1 - 7.15T + 47T^{2} \)
53 \( 1 + 9.05T + 53T^{2} \)
59 \( 1 - 6.48iT - 59T^{2} \)
61 \( 1 + 1.48iT - 61T^{2} \)
67 \( 1 + 13.9T + 67T^{2} \)
71 \( 1 - 2.52T + 71T^{2} \)
73 \( 1 - 5.63T + 73T^{2} \)
79 \( 1 + 6.77iT - 79T^{2} \)
83 \( 1 - 14.6iT - 83T^{2} \)
89 \( 1 + 6.73iT - 89T^{2} \)
97 \( 1 + 3.34T + 97T^{2} \)
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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

−10.39373601762932347615698087154, −9.859048197993910350574065847639, −8.612808541641886753526920490000, −7.83526869271780277605315643411, −6.71450397545037975584712442462, −6.23708983720908753308972550682, −4.67669579350199938016350021025, −4.22792449261609558978178324194, −3.46281265445240931795599005945, −1.35376000513481228759975009818, 0.32581222746375736969159998301, 2.16460412778715340718002778936, 2.99334970229130652985588089355, 4.72050408109971312484830045814, 5.53809974939559969517812749192, 6.16663616103188858835083725241, 7.19052699745288473903418492295, 8.080725308158701635638336652708, 8.727609171346884235874386757105, 9.670502911182542831648913828039

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