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Properties

Label 2-960-24.11-c1-0-10
Degree $2$
Conductor $960$
Sign $0.814 - 0.580i$
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  + (1.10 − 1.33i)3-s − 5-s + 4.67i·7-s + (−0.571 − 2.94i)9-s − 1.14i·11-s + 2.42i·13-s + (−1.10 + 1.33i)15-s + 7.87i·17-s + 2.42·19-s + (6.24 + 5.14i)21-s + 4.62·23-s + 25-s + (−4.56 − 2.48i)27-s + 6.48·29-s − 3.14i·31-s + ⋯
L(s)  = 1  + (0.636 − 0.771i)3-s − 0.447·5-s + 1.76i·7-s + (−0.190 − 0.981i)9-s − 0.344i·11-s + 0.672i·13-s + (−0.284 + 0.345i)15-s + 1.90i·17-s + 0.556·19-s + (1.36 + 1.12i)21-s + 0.965·23-s + 0.200·25-s + (−0.878 − 0.477i)27-s + 1.20·29-s − 0.564i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.814 - 0.580i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.814 - 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.814 - 0.580i$
Analytic conductor: \(7.66563\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (671, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ 0.814 - 0.580i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.63978 + 0.524833i\)
\(L(\frac12)\) \(\approx\) \(1.63978 + 0.524833i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.10 + 1.33i)T \)
5 \( 1 + T \)
good7 \( 1 - 4.67iT - 7T^{2} \)
11 \( 1 + 1.14iT - 11T^{2} \)
13 \( 1 - 2.42iT - 13T^{2} \)
17 \( 1 - 7.87iT - 17T^{2} \)
19 \( 1 - 2.42T + 19T^{2} \)
23 \( 1 - 4.62T + 23T^{2} \)
29 \( 1 - 6.48T + 29T^{2} \)
31 \( 1 + 3.14iT - 31T^{2} \)
37 \( 1 - 6.83iT - 37T^{2} \)
41 \( 1 - 5.88iT - 41T^{2} \)
43 \( 1 - 6.61T + 43T^{2} \)
47 \( 1 + 7.15T + 47T^{2} \)
53 \( 1 + 9.05T + 53T^{2} \)
59 \( 1 + 6.48iT - 59T^{2} \)
61 \( 1 + 1.48iT - 61T^{2} \)
67 \( 1 - 13.9T + 67T^{2} \)
71 \( 1 + 2.52T + 71T^{2} \)
73 \( 1 - 5.63T + 73T^{2} \)
79 \( 1 - 6.77iT - 79T^{2} \)
83 \( 1 + 14.6iT - 83T^{2} \)
89 \( 1 + 6.73iT - 89T^{2} \)
97 \( 1 + 3.34T + 97T^{2} \)
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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

−9.849954764861434269935991913580, −9.008210717760073272626911900246, −8.413124565461366104169271212738, −7.88804265204399318637942325348, −6.53182903696575202490782023941, −6.11604185759055648433549395354, −4.88593947595531243711842778457, −3.51409346300508511101770020899, −2.63524009913064636499463783307, −1.52476178123500164017694803561, 0.802206337045275222921611076542, 2.80331029359383192487182976789, 3.61508486679679324889242815480, 4.56746296537471933045889206948, 5.18305012621651765573344617780, 6.97081902847398825154966372084, 7.40578930746129068951638398377, 8.199878992611713405642079744194, 9.305309853730170175376735991647, 9.896201270888429771517239298388

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