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Properties

Label 2-960-24.11-c1-0-0
Degree $2$
Conductor $960$
Sign $-0.990 + 0.138i$
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  + (−1.59 + 0.675i)3-s − 5-s + 0.648i·7-s + (2.08 − 2.15i)9-s + 4.17i·11-s + 0.847i·13-s + (1.59 − 0.675i)15-s − 2.91i·17-s + 0.847·19-s + (−0.438 − 1.03i)21-s − 2.34·23-s + 25-s + (−1.86 + 4.84i)27-s − 6.87·29-s + 2.17i·31-s + ⋯
L(s)  = 1  + (−0.920 + 0.390i)3-s − 0.447·5-s + 0.244i·7-s + (0.695 − 0.718i)9-s + 1.25i·11-s + 0.235i·13-s + (0.411 − 0.174i)15-s − 0.706i·17-s + 0.194·19-s + (−0.0955 − 0.225i)21-s − 0.488·23-s + 0.200·25-s + (−0.359 + 0.933i)27-s − 1.27·29-s + 0.390i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.138i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-0.990 + 0.138i$
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: \((2,\ 960,\ (\ :1/2),\ -0.990 + 0.138i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0151897 - 0.218005i\)
\(L(\frac12)\) \(\approx\) \(0.0151897 - 0.218005i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.59 - 0.675i)T \)
5 \( 1 + T \)
good7 \( 1 - 0.648iT - 7T^{2} \)
11 \( 1 - 4.17iT - 11T^{2} \)
13 \( 1 - 0.847iT - 13T^{2} \)
17 \( 1 + 2.91iT - 17T^{2} \)
19 \( 1 - 0.847T + 19T^{2} \)
23 \( 1 + 2.34T + 23T^{2} \)
29 \( 1 + 6.87T + 29T^{2} \)
31 \( 1 - 2.17iT - 31T^{2} \)
37 \( 1 + 5.53iT - 37T^{2} \)
41 \( 1 - 4.31iT - 41T^{2} \)
43 \( 1 + 9.56T + 43T^{2} \)
47 \( 1 + 10.9T + 47T^{2} \)
53 \( 1 + 11.6T + 53T^{2} \)
59 \( 1 - 6.87iT - 59T^{2} \)
61 \( 1 + 10.6iT - 61T^{2} \)
67 \( 1 - 5.43T + 67T^{2} \)
71 \( 1 + 13.3T + 71T^{2} \)
73 \( 1 + 13.0T + 73T^{2} \)
79 \( 1 + 17.2iT - 79T^{2} \)
83 \( 1 + 10.6iT - 83T^{2} \)
89 \( 1 - 17.9iT - 89T^{2} \)
97 \( 1 - 4.70T + 97T^{2} \)
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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

−10.43445682779511445094507152257, −9.703150443266500623581818084691, −9.056884222073544490570507827177, −7.73923698562446048586827909602, −7.06256970427518972272589038852, −6.16809104450113849274991058759, −5.09294753166696519418780787159, −4.48704247289809108900869017231, −3.39359179514388774226556432152, −1.76228189338800663953414840703, 0.11611367226232990760487100330, 1.55323459096572091379480661676, 3.25130392139149742140500826750, 4.26800360381303907943846911666, 5.40762200318289301535335340417, 6.09106916256253837040608937435, 6.97550978399104988593814907841, 7.905562777875721032275556894196, 8.519652467105170555161995810747, 9.806173869303934728766802006827

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