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Properties

Label 2-960-24.11-c3-0-13
Degree $2$
Conductor $960$
Sign $-0.999 - 0.0159i$
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  + (3.61 + 3.73i)3-s + 5·5-s + 29.1i·7-s + (−0.862 + 26.9i)9-s − 37.6i·11-s − 58.2i·13-s + (18.0 + 18.6i)15-s + 73.3i·17-s − 95.1·19-s + (−108. + 105. i)21-s − 189.·23-s + 25·25-s + (−103. + 94.3i)27-s + 68.9·29-s + 264. i·31-s + ⋯
L(s)  = 1  + (0.695 + 0.718i)3-s + 0.447·5-s + 1.57i·7-s + (−0.0319 + 0.999i)9-s − 1.03i·11-s − 1.24i·13-s + (0.311 + 0.321i)15-s + 1.04i·17-s − 1.14·19-s + (−1.13 + 1.09i)21-s − 1.72·23-s + 0.200·25-s + (−0.740 + 0.672i)27-s + 0.441·29-s + 1.53i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0159i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0159i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-0.999 - 0.0159i$
Analytic conductor: \(56.6418\)
Root analytic conductor: \(7.52607\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (671, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :3/2),\ -0.999 - 0.0159i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.506851822\)
\(L(\frac12)\) \(\approx\) \(1.506851822\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.61 - 3.73i)T \)
5 \( 1 - 5T \)
good7 \( 1 - 29.1iT - 343T^{2} \)
11 \( 1 + 37.6iT - 1.33e3T^{2} \)
13 \( 1 + 58.2iT - 2.19e3T^{2} \)
17 \( 1 - 73.3iT - 4.91e3T^{2} \)
19 \( 1 + 95.1T + 6.85e3T^{2} \)
23 \( 1 + 189.T + 1.21e4T^{2} \)
29 \( 1 - 68.9T + 2.43e4T^{2} \)
31 \( 1 - 264. iT - 2.97e4T^{2} \)
37 \( 1 - 338. iT - 5.06e4T^{2} \)
41 \( 1 + 145. iT - 6.89e4T^{2} \)
43 \( 1 + 51.9T + 7.95e4T^{2} \)
47 \( 1 + 173.T + 1.03e5T^{2} \)
53 \( 1 - 212.T + 1.48e5T^{2} \)
59 \( 1 + 699. iT - 2.05e5T^{2} \)
61 \( 1 - 292. iT - 2.26e5T^{2} \)
67 \( 1 + 919.T + 3.00e5T^{2} \)
71 \( 1 - 563.T + 3.57e5T^{2} \)
73 \( 1 + 11.2T + 3.89e5T^{2} \)
79 \( 1 - 680. iT - 4.93e5T^{2} \)
83 \( 1 + 696. iT - 5.71e5T^{2} \)
89 \( 1 + 1.64e3iT - 7.04e5T^{2} \)
97 \( 1 + 437.T + 9.12e5T^{2} \)
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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

−10.21125488860315244937157217284, −9.041038573800162832041148683429, −8.391093636513222076446908087745, −8.142160308822461397553845980671, −6.31667740219769161595532100801, −5.74702051948455585336677307242, −4.89946504811180562258810703530, −3.57682885958326854348602605245, −2.77185807983605420315793285482, −1.82524176272505236250445245900, 0.30705335429984263348087330347, 1.65128619543445664016800485411, 2.40816526545133898897937674873, 3.99872456571999393800814452250, 4.40056228248010923041117845510, 6.08949462710938116548485799934, 6.90994222789457060507435739168, 7.39906595958231244694620134687, 8.247805692812581094896998729109, 9.419868032531057935245703291298

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