Properties

Label 2-960-5.4-c3-0-38
Degree $2$
Conductor $960$
Sign $-0.116 + 0.993i$
Analytic cond. $56.6418$
Root an. cond. $7.52607$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3i·3-s + (−11.1 − 1.29i)5-s + 16.2i·7-s − 9·9-s − 40.2·11-s + 19.7i·13-s + (−3.89 + 33.3i)15-s + 83.0i·17-s + 48.8·19-s + 48.6·21-s + 1.61i·23-s + (121. + 28.8i)25-s + 27i·27-s − 24.5·29-s + 12.4·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (−0.993 − 0.116i)5-s + 0.875i·7-s − 0.333·9-s − 1.10·11-s + 0.422i·13-s + (−0.0670 + 0.573i)15-s + 1.18i·17-s + 0.589·19-s + 0.505·21-s + 0.0146i·23-s + (0.973 + 0.230i)25-s + 0.192i·27-s − 0.157·29-s + 0.0719·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.116 + 0.993i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.116 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-0.116 + 0.993i$
Analytic conductor: \(56.6418\)
Root analytic conductor: \(7.52607\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :3/2),\ -0.116 + 0.993i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7398836158\)
\(L(\frac12)\) \(\approx\) \(0.7398836158\)
\(L(\frac{5}{2})\) 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 \( 1 + 3iT \)
5 \( 1 + (11.1 + 1.29i)T \)
good7 \( 1 - 16.2iT - 343T^{2} \)
11 \( 1 + 40.2T + 1.33e3T^{2} \)
13 \( 1 - 19.7iT - 2.19e3T^{2} \)
17 \( 1 - 83.0iT - 4.91e3T^{2} \)
19 \( 1 - 48.8T + 6.85e3T^{2} \)
23 \( 1 - 1.61iT - 1.21e4T^{2} \)
29 \( 1 + 24.5T + 2.43e4T^{2} \)
31 \( 1 - 12.4T + 2.97e4T^{2} \)
37 \( 1 + 325. iT - 5.06e4T^{2} \)
41 \( 1 + 242.T + 6.89e4T^{2} \)
43 \( 1 + 367. iT - 7.95e4T^{2} \)
47 \( 1 - 204. iT - 1.03e5T^{2} \)
53 \( 1 + 61.5iT - 1.48e5T^{2} \)
59 \( 1 - 112.T + 2.05e5T^{2} \)
61 \( 1 + 477.T + 2.26e5T^{2} \)
67 \( 1 - 558. iT - 3.00e5T^{2} \)
71 \( 1 + 558.T + 3.57e5T^{2} \)
73 \( 1 + 1.01e3iT - 3.89e5T^{2} \)
79 \( 1 - 1.15e3T + 4.93e5T^{2} \)
83 \( 1 - 1.15e3iT - 5.71e5T^{2} \)
89 \( 1 + 96.9T + 7.04e5T^{2} \)
97 \( 1 + 1.15e3iT - 9.12e5T^{2} \)
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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

−9.163300974027634675921210679798, −8.468444708679054963411658983035, −7.78516268670009788251395143883, −7.04579010513617666373750870874, −5.91794218707413854285266948417, −5.16172532638493311697288665766, −3.97110719126549774277080327935, −2.89842614835947036478143944502, −1.80484286687918247020763659406, −0.25631775832331570205085822254, 0.810534400152199493417260728516, 2.80682396021086534671052688076, 3.52424259675361632723343513281, 4.63897666143510660975661492301, 5.19805131087996443757181601055, 6.60461857970188157464548023136, 7.57654437412044950683577284029, 7.975465943201648850639631464684, 9.063668819743665929497923799160, 10.08845511056450179096439683863

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