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Properties

Label 2-960-24.11-c1-0-17
Degree $2$
Conductor $960$
Sign $0.999 + 0.0267i$
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  + (−0.492 + 1.66i)3-s − 5-s − 1.32i·7-s + (−2.51 − 1.63i)9-s − 5.02i·11-s + 6.73i·13-s + (0.492 − 1.66i)15-s − 5.43i·17-s + 6.73·19-s + (2.19 + 0.651i)21-s + 5.75·23-s + 25-s + (3.95 − 3.36i)27-s − 1.61·29-s − 7.02i·31-s + ⋯
L(s)  = 1  + (−0.284 + 0.958i)3-s − 0.447·5-s − 0.499i·7-s + (−0.838 − 0.545i)9-s − 1.51i·11-s + 1.86i·13-s + (0.127 − 0.428i)15-s − 1.31i·17-s + 1.54·19-s + (0.478 + 0.142i)21-s + 1.19·23-s + 0.200·25-s + (0.761 − 0.648i)27-s − 0.299·29-s − 1.26i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0267i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0267i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.999 + 0.0267i$
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: \((2,\ 960,\ (\ :1/2),\ 0.999 + 0.0267i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24106 - 0.0166022i\)
\(L(\frac12)\) \(\approx\) \(1.24106 - 0.0166022i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.492 - 1.66i)T \)
5 \( 1 + T \)
good7 \( 1 + 1.32iT - 7T^{2} \)
11 \( 1 + 5.02iT - 11T^{2} \)
13 \( 1 - 6.73iT - 13T^{2} \)
17 \( 1 + 5.43iT - 17T^{2} \)
19 \( 1 - 6.73T + 19T^{2} \)
23 \( 1 - 5.75T + 23T^{2} \)
29 \( 1 + 1.61T + 29T^{2} \)
31 \( 1 + 7.02iT - 31T^{2} \)
37 \( 1 - 4.76iT - 37T^{2} \)
41 \( 1 - 3.27iT - 41T^{2} \)
43 \( 1 + 2.95T + 43T^{2} \)
47 \( 1 + 0.795T + 47T^{2} \)
53 \( 1 - 10.6T + 53T^{2} \)
59 \( 1 - 1.61iT - 59T^{2} \)
61 \( 1 + 5.24iT - 61T^{2} \)
67 \( 1 - 5.56T + 67T^{2} \)
71 \( 1 - 4.95T + 71T^{2} \)
73 \( 1 - 1.41T + 73T^{2} \)
79 \( 1 - 6.44iT - 79T^{2} \)
83 \( 1 + 8.67iT - 83T^{2} \)
89 \( 1 + 16.4iT - 89T^{2} \)
97 \( 1 - 8.64T + 97T^{2} \)
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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

−9.895394937654590442482575307364, −9.268776026640275947045373542758, −8.615696568836205636899603288157, −7.42929463431068348436939699925, −6.62001299834210009081116984119, −5.52231288318084650557855170243, −4.68198177966655569521827502947, −3.76567373738903166596912433032, −2.93366950448713220454952376883, −0.75804052334874974227723876057, 1.10850982366940730972033120221, 2.45500238436513593033763886399, 3.53231073690860092856499855370, 5.12782204398199375243021855362, 5.56735829898205818406547675809, 6.84270422788753516988036088337, 7.47591568634270147745145009375, 8.152193943145729485321899696285, 9.043963100940011747835070284867, 10.21094601715172106932393772829

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