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Properties

Label 2-960-20.3-c1-0-8
Degree $2$
Conductor $960$
Sign $0.619 - 0.784i$
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 + (1.75 − 1.38i)5-s + (2.47 + 2.47i)7-s − 1.00i·9-s + 3.02i·11-s + (−0.363 − 0.363i)13-s + (−0.256 + 2.22i)15-s + (−2.36 + 2.36i)17-s + 4.95·19-s − 3.50·21-s + (−0.900 + 0.900i)23-s + (1.14 − 4.86i)25-s + (0.707 + 0.707i)27-s + 3.50i·29-s − 3.85i·31-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (0.783 − 0.621i)5-s + (0.936 + 0.936i)7-s − 0.333i·9-s + 0.913i·11-s + (−0.100 − 0.100i)13-s + (−0.0663 + 0.573i)15-s + (−0.573 + 0.573i)17-s + 1.13·19-s − 0.764·21-s + (−0.187 + 0.187i)23-s + (0.228 − 0.973i)25-s + (0.136 + 0.136i)27-s + 0.650i·29-s − 0.692i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.619 - 0.784i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.619 - 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.619 - 0.784i$
Analytic conductor: \(7.66563\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ 0.619 - 0.784i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55795 + 0.754572i\)
\(L(\frac12)\) \(\approx\) \(1.55795 + 0.754572i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.75 + 1.38i)T \)
good7 \( 1 + (-2.47 - 2.47i)T + 7iT^{2} \)
11 \( 1 - 3.02iT - 11T^{2} \)
13 \( 1 + (0.363 + 0.363i)T + 13iT^{2} \)
17 \( 1 + (2.36 - 2.36i)T - 17iT^{2} \)
19 \( 1 - 4.95T + 19T^{2} \)
23 \( 1 + (0.900 - 0.900i)T - 23iT^{2} \)
29 \( 1 - 3.50iT - 29T^{2} \)
31 \( 1 + 3.85iT - 31T^{2} \)
37 \( 1 + (-0.363 + 0.363i)T - 37iT^{2} \)
41 \( 1 - 2.72T + 41T^{2} \)
43 \( 1 + (3.92 - 3.92i)T - 43iT^{2} \)
47 \( 1 + (-5.85 - 5.85i)T + 47iT^{2} \)
53 \( 1 + (3.14 + 3.14i)T + 53iT^{2} \)
59 \( 1 + 8.68T + 59T^{2} \)
61 \( 1 - 15.2T + 61T^{2} \)
67 \( 1 + (-3.92 - 3.92i)T + 67iT^{2} \)
71 \( 1 - 4.25iT - 71T^{2} \)
73 \( 1 + (-9.28 - 9.28i)T + 73iT^{2} \)
79 \( 1 - 0.399T + 79T^{2} \)
83 \( 1 + (-0.199 + 0.199i)T - 83iT^{2} \)
89 \( 1 + 4.28iT - 89T^{2} \)
97 \( 1 + (-6.73 + 6.73i)T - 97iT^{2} \)
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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

−9.963558745237338188686421407520, −9.417165975470359699391491661611, −8.628079166161726908680549998990, −7.76273257866395863661072044309, −6.56025664421716493626098444579, −5.54308032175776171099056655294, −5.08908486628400419052136971170, −4.17296348945432152472847547948, −2.49528478668571147048569068329, −1.46578154673600151311161677004, 0.953912679348357118950716059191, 2.21690203767037814300871394571, 3.48857817310581709064671377257, 4.79987669165103069684328225517, 5.60395165043796984973904665105, 6.57054747616350908097917285867, 7.28209846080838010205513971156, 8.058898011223107083019350008671, 9.131418072113289204948382710183, 10.08095610757754581187530650277

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