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Properties

Label 2-960-20.7-c1-0-2
Degree $2$
Conductor $960$
Sign $0.0372 - 0.999i$
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 + (−0.432 − 2.19i)5-s + (−0.611 + 0.611i)7-s + 1.00i·9-s + 5.12i·11-s + (−1.76 + 1.76i)13-s + (−1.24 + 1.85i)15-s + (−3.76 − 3.76i)17-s − 1.22·19-s + 0.864·21-s + (1.07 + 1.07i)23-s + (−4.62 + 1.89i)25-s + (0.707 − 0.707i)27-s + 0.864i·29-s + 7.81i·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (−0.193 − 0.981i)5-s + (−0.231 + 0.231i)7-s + 0.333i·9-s + 1.54i·11-s + (−0.488 + 0.488i)13-s + (−0.321 + 0.479i)15-s + (−0.912 − 0.912i)17-s − 0.280·19-s + 0.188·21-s + (0.224 + 0.224i)23-s + (−0.925 + 0.379i)25-s + (0.136 − 0.136i)27-s + 0.160i·29-s + 1.40i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0372 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0372 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.0372 - 0.999i$
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: \((2,\ 960,\ (\ :1/2),\ 0.0372 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.448412 + 0.432008i\)
\(L(\frac12)\) \(\approx\) \(0.448412 + 0.432008i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.432 + 2.19i)T \)
good7 \( 1 + (0.611 - 0.611i)T - 7iT^{2} \)
11 \( 1 - 5.12iT - 11T^{2} \)
13 \( 1 + (1.76 - 1.76i)T - 13iT^{2} \)
17 \( 1 + (3.76 + 3.76i)T + 17iT^{2} \)
19 \( 1 + 1.22T + 19T^{2} \)
23 \( 1 + (-1.07 - 1.07i)T + 23iT^{2} \)
29 \( 1 - 0.864iT - 29T^{2} \)
31 \( 1 - 7.81iT - 31T^{2} \)
37 \( 1 + (-1.76 - 1.76i)T + 37iT^{2} \)
41 \( 1 - 5.52T + 41T^{2} \)
43 \( 1 + (-6.20 - 6.20i)T + 43iT^{2} \)
47 \( 1 + (2.29 - 2.29i)T - 47iT^{2} \)
53 \( 1 + (-2.62 + 2.62i)T - 53iT^{2} \)
59 \( 1 + 0.528T + 59T^{2} \)
61 \( 1 + 4.98T + 61T^{2} \)
67 \( 1 + (6.20 - 6.20i)T - 67iT^{2} \)
71 \( 1 - 8.10iT - 71T^{2} \)
73 \( 1 + (2.25 - 2.25i)T - 73iT^{2} \)
79 \( 1 + 15.9T + 79T^{2} \)
83 \( 1 + (7.95 + 7.95i)T + 83iT^{2} \)
89 \( 1 + 7.25iT - 89T^{2} \)
97 \( 1 + (-0.793 - 0.793i)T + 97iT^{2} \)
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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

−10.07853397209684561563300533645, −9.344314572597450766311489886630, −8.700425414901049959687588045523, −7.48881221207135058156181559123, −7.01688763204191926048364462803, −5.91137736471089483856343931987, −4.76615341256557184563305089796, −4.44378714572351436019577403612, −2.61419892947092792426643221491, −1.44831096677768928374013255299, 0.30636422141809643507542628790, 2.47209937612476165362769705769, 3.51681833112377217499989454731, 4.31610728291258210420332056892, 5.76949137725950232503710765440, 6.20880002314003126240268110179, 7.22231333705834450619650698705, 8.144790692297455210678506214269, 9.033356523257315426734714844594, 10.04697810013841842487179867401

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