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Properties

Label 2-960-3.2-c2-0-57
Degree $2$
Conductor $960$
Sign $0.288 + 0.957i$
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  + (2.87 − 0.864i)3-s + 2.23i·5-s + 9.02·7-s + (7.50 − 4.96i)9-s − 21.8i·11-s − 21.6·13-s + (1.93 + 6.42i)15-s − 12.1i·17-s − 3.03·19-s + (25.9 − 7.80i)21-s − 28.5i·23-s − 5.00·25-s + (17.2 − 20.7i)27-s − 12.0i·29-s + 2.19·31-s + ⋯
L(s)  = 1  + (0.957 − 0.288i)3-s + 0.447i·5-s + 1.28·7-s + (0.833 − 0.551i)9-s − 1.98i·11-s − 1.66·13-s + (0.128 + 0.428i)15-s − 0.712i·17-s − 0.159·19-s + (1.23 − 0.371i)21-s − 1.24i·23-s − 0.200·25-s + (0.639 − 0.768i)27-s − 0.415i·29-s + 0.0706·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.288 + 0.957i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.288 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.288 + 0.957i$
Analytic conductor: \(26.1581\)
Root analytic conductor: \(5.11449\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1),\ 0.288 + 0.957i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.788780533\)
\(L(\frac12)\) \(\approx\) \(2.788780533\)
\(L(2)\) 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 \( 1 + (-2.87 + 0.864i)T \)
5 \( 1 - 2.23iT \)
good7 \( 1 - 9.02T + 49T^{2} \)
11 \( 1 + 21.8iT - 121T^{2} \)
13 \( 1 + 21.6T + 169T^{2} \)
17 \( 1 + 12.1iT - 289T^{2} \)
19 \( 1 + 3.03T + 361T^{2} \)
23 \( 1 + 28.5iT - 529T^{2} \)
29 \( 1 + 12.0iT - 841T^{2} \)
31 \( 1 - 2.19T + 961T^{2} \)
37 \( 1 + 0.839T + 1.36e3T^{2} \)
41 \( 1 - 35.5iT - 1.68e3T^{2} \)
43 \( 1 - 12.7T + 1.84e3T^{2} \)
47 \( 1 - 22.5iT - 2.20e3T^{2} \)
53 \( 1 + 9.13iT - 2.80e3T^{2} \)
59 \( 1 - 80.4iT - 3.48e3T^{2} \)
61 \( 1 - 57.8T + 3.72e3T^{2} \)
67 \( 1 - 63.0T + 4.48e3T^{2} \)
71 \( 1 - 17.0iT - 5.04e3T^{2} \)
73 \( 1 - 52.1T + 5.32e3T^{2} \)
79 \( 1 + 7.46T + 6.24e3T^{2} \)
83 \( 1 - 82.3iT - 6.88e3T^{2} \)
89 \( 1 + 27.5iT - 7.92e3T^{2} \)
97 \( 1 - 114.T + 9.40e3T^{2} \)
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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

−9.557636099137463636172590246766, −8.608714393697428648695826069978, −8.076444649269275989410430976473, −7.34661115042536846124180547898, −6.38958637449575876182947254612, −5.19869840407602031490658478980, −4.24516066848529849155090113877, −2.97868931703802910700287135848, −2.30015932239001820584345431641, −0.77374425550967042959609197277, 1.72670517189078480305629408677, 2.23601257501615640023003875025, 3.87010146637927527485421533639, 4.77895279971463758987197991596, 5.14947704414428872336331039153, 7.05307077363023842668467925063, 7.58247024732033315374752622719, 8.246691826364871487654061315095, 9.281618139643172370408054895845, 9.820456678886109877438595991431

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