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Properties

Label 2-960-5.3-c2-0-15
Degree $2$
Conductor $960$
Sign $0.130 - 0.991i$
Analytic cond. $26.1581$
Root an. cond. $5.11449$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 − 1.22i)3-s + (4.67 + 1.77i)5-s + (−3.44 + 3.44i)7-s + 2.99i·9-s + 11.3·11-s + (5.55 + 5.55i)13-s + (−3.55 − 7.89i)15-s + (−17.3 + 17.3i)17-s − 8.69i·19-s + 8.44·21-s + (−11.5 − 11.5i)23-s + (18.6 + 16.5i)25-s + (3.67 − 3.67i)27-s + 35.1i·29-s − 10.6·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (0.934 + 0.355i)5-s + (−0.492 + 0.492i)7-s + 0.333i·9-s + 1.03·11-s + (0.426 + 0.426i)13-s + (−0.236 − 0.526i)15-s + (−1.02 + 1.02i)17-s − 0.457i·19-s + 0.402·21-s + (−0.502 − 0.502i)23-s + (0.747 + 0.663i)25-s + (0.136 − 0.136i)27-s + 1.21i·29-s − 0.345·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.130 - 0.991i$
Analytic conductor: \(26.1581\)
Root analytic conductor: \(5.11449\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1),\ 0.130 - 0.991i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.535884803\)
\(L(\frac12)\) \(\approx\) \(1.535884803\)
\(L(2)\) 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 \( 1 + (1.22 + 1.22i)T \)
5 \( 1 + (-4.67 - 1.77i)T \)
good7 \( 1 + (3.44 - 3.44i)T - 49iT^{2} \)
11 \( 1 - 11.3T + 121T^{2} \)
13 \( 1 + (-5.55 - 5.55i)T + 169iT^{2} \)
17 \( 1 + (17.3 - 17.3i)T - 289iT^{2} \)
19 \( 1 + 8.69iT - 361T^{2} \)
23 \( 1 + (11.5 + 11.5i)T + 529iT^{2} \)
29 \( 1 - 35.1iT - 841T^{2} \)
31 \( 1 + 10.6T + 961T^{2} \)
37 \( 1 + (-6.04 + 6.04i)T - 1.36e3iT^{2} \)
41 \( 1 - 0.696T + 1.68e3T^{2} \)
43 \( 1 + (26.4 + 26.4i)T + 1.84e3iT^{2} \)
47 \( 1 + (44.2 - 44.2i)T - 2.20e3iT^{2} \)
53 \( 1 + (-0.696 - 0.696i)T + 2.80e3iT^{2} \)
59 \( 1 - 39.9iT - 3.48e3T^{2} \)
61 \( 1 + 5.90T + 3.72e3T^{2} \)
67 \( 1 + (45.1 - 45.1i)T - 4.48e3iT^{2} \)
71 \( 1 - 68T + 5.04e3T^{2} \)
73 \( 1 + (-77.7 - 77.7i)T + 5.32e3iT^{2} \)
79 \( 1 + 24.4iT - 6.24e3T^{2} \)
83 \( 1 + (-13.1 - 13.1i)T + 6.88e3iT^{2} \)
89 \( 1 + 82.1iT - 7.92e3T^{2} \)
97 \( 1 + (24.5 - 24.5i)T - 9.40e3iT^{2} \)
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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

−10.04125518058919595025756413024, −9.083376214471879825759490002845, −8.650173260971409383775361502239, −7.18554568399562725961552908337, −6.34883936889860658813193998609, −6.13148910832198930810577160037, −4.88697070471680446458852867765, −3.67342863018173567163842059543, −2.36320560673298059042063974620, −1.41460677925966224785878232625, 0.52155308691460772439941743620, 1.88547945409243602101623621295, 3.36775021217720187791989828562, 4.34603818690796812411319047415, 5.30176155245537527868827589047, 6.27090819703892398078911297482, 6.74958544346448284937424732896, 8.061924059902275882311824451343, 9.122773510520820998787906589900, 9.648275063931674626310998370534

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