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Properties

Label 2-960-20.7-c1-0-4
Degree $2$
Conductor $960$
Sign $-0.899 - 0.437i$
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  + (0.707 + 0.707i)3-s + (−0.489 + 2.18i)5-s + (−0.692 + 0.692i)7-s + 1.00i·9-s + 0.979i·11-s + (−3.49 + 3.49i)13-s + (−1.88 + 1.19i)15-s + (−2.67 − 2.67i)17-s + 3.34·19-s − 0.979·21-s + (−3.80 − 3.80i)23-s + (−4.52 − 2.13i)25-s + (−0.707 + 0.707i)27-s + 3.74i·29-s + 4i·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (−0.218 + 0.975i)5-s + (−0.261 + 0.261i)7-s + 0.333i·9-s + 0.295i·11-s + (−0.970 + 0.970i)13-s + (−0.487 + 0.308i)15-s + (−0.647 − 0.647i)17-s + 0.766·19-s − 0.213·21-s + (−0.793 − 0.793i)23-s + (−0.904 − 0.427i)25-s + (−0.136 + 0.136i)27-s + 0.696i·29-s + 0.718i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.899 - 0.437i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.899 - 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-0.899 - 0.437i$
Analytic conductor: \(7.66563\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ -0.899 - 0.437i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.241133 + 1.04745i\)
\(L(\frac12)\) \(\approx\) \(0.241133 + 1.04745i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (0.489 - 2.18i)T \)
good7 \( 1 + (0.692 - 0.692i)T - 7iT^{2} \)
11 \( 1 - 0.979iT - 11T^{2} \)
13 \( 1 + (3.49 - 3.49i)T - 13iT^{2} \)
17 \( 1 + (2.67 + 2.67i)T + 17iT^{2} \)
19 \( 1 - 3.34T + 19T^{2} \)
23 \( 1 + (3.80 + 3.80i)T + 23iT^{2} \)
29 \( 1 - 3.74iT - 29T^{2} \)
31 \( 1 - 4iT - 31T^{2} \)
37 \( 1 + (4.11 + 4.11i)T + 37iT^{2} \)
41 \( 1 + 3.34T + 41T^{2} \)
43 \( 1 + (-8.21 - 8.21i)T + 43iT^{2} \)
47 \( 1 + (-9.19 + 9.19i)T - 47iT^{2} \)
53 \( 1 + (7.34 - 7.34i)T - 53iT^{2} \)
59 \( 1 + 11.3T + 59T^{2} \)
61 \( 1 + 1.68T + 61T^{2} \)
67 \( 1 + (-4.51 + 4.51i)T - 67iT^{2} \)
71 \( 1 - 5.65iT - 71T^{2} \)
73 \( 1 + (6.34 - 6.34i)T - 73iT^{2} \)
79 \( 1 - 16.3T + 79T^{2} \)
83 \( 1 + (4.91 + 4.91i)T + 83iT^{2} \)
89 \( 1 - 12.6iT - 89T^{2} \)
97 \( 1 + (-6.38 - 6.38i)T + 97iT^{2} \)
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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

−10.35602714572487020209217958250, −9.487618455469444510078292169020, −8.970133904857321543436862355530, −7.70811464512400583341889790871, −7.09730545231708826423991195701, −6.26804303203317932239052736942, −4.98915553513101434659995378836, −4.10869803107703551251500265463, −2.98456462461537860111860472508, −2.18147062189300053246221555894, 0.44874906152874954872764080781, 1.91859151830482869388918488735, 3.25590854521229980881805712197, 4.25598821670320177750176298891, 5.33271967963995559407952306434, 6.16868069353769778178165174728, 7.47574482251521427865830322329, 7.87916994497391140027590366070, 8.806141776581272841213841139244, 9.577440928128243345534759624903

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