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Properties

Label 2-960-5.3-c2-0-20
Degree $2$
Conductor $960$
Sign $0.130 - 0.991i$
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  + (1.22 + 1.22i)3-s + (4.67 + 1.77i)5-s + (3.44 − 3.44i)7-s + 2.99i·9-s − 11.3·11-s + (5.55 + 5.55i)13-s + (3.55 + 7.89i)15-s + (−17.3 + 17.3i)17-s + 8.69i·19-s + 8.44·21-s + (11.5 + 11.5i)23-s + (18.6 + 16.5i)25-s + (−3.67 + 3.67i)27-s + 35.1i·29-s + 10.6·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (0.934 + 0.355i)5-s + (0.492 − 0.492i)7-s + 0.333i·9-s − 1.03·11-s + (0.426 + 0.426i)13-s + (0.236 + 0.526i)15-s + (−1.02 + 1.02i)17-s + 0.457i·19-s + 0.402·21-s + (0.502 + 0.502i)23-s + (0.747 + 0.663i)25-s + (−0.136 + 0.136i)27-s + 1.21i·29-s + 0.345·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.130 - 0.991i$
Analytic conductor: \(26.1581\)
Root analytic conductor: \(5.11449\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1),\ 0.130 - 0.991i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.449023725\)
\(L(\frac12)\) \(\approx\) \(2.449023725\)
\(L(2)\) 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 \( 1 + (-1.22 - 1.22i)T \)
5 \( 1 + (-4.67 - 1.77i)T \)
good7 \( 1 + (-3.44 + 3.44i)T - 49iT^{2} \)
11 \( 1 + 11.3T + 121T^{2} \)
13 \( 1 + (-5.55 - 5.55i)T + 169iT^{2} \)
17 \( 1 + (17.3 - 17.3i)T - 289iT^{2} \)
19 \( 1 - 8.69iT - 361T^{2} \)
23 \( 1 + (-11.5 - 11.5i)T + 529iT^{2} \)
29 \( 1 - 35.1iT - 841T^{2} \)
31 \( 1 - 10.6T + 961T^{2} \)
37 \( 1 + (-6.04 + 6.04i)T - 1.36e3iT^{2} \)
41 \( 1 - 0.696T + 1.68e3T^{2} \)
43 \( 1 + (-26.4 - 26.4i)T + 1.84e3iT^{2} \)
47 \( 1 + (-44.2 + 44.2i)T - 2.20e3iT^{2} \)
53 \( 1 + (-0.696 - 0.696i)T + 2.80e3iT^{2} \)
59 \( 1 + 39.9iT - 3.48e3T^{2} \)
61 \( 1 + 5.90T + 3.72e3T^{2} \)
67 \( 1 + (-45.1 + 45.1i)T - 4.48e3iT^{2} \)
71 \( 1 + 68T + 5.04e3T^{2} \)
73 \( 1 + (-77.7 - 77.7i)T + 5.32e3iT^{2} \)
79 \( 1 - 24.4iT - 6.24e3T^{2} \)
83 \( 1 + (13.1 + 13.1i)T + 6.88e3iT^{2} \)
89 \( 1 + 82.1iT - 7.92e3T^{2} \)
97 \( 1 + (24.5 - 24.5i)T - 9.40e3iT^{2} \)
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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

−10.12970969084998704124095958282, −9.184317258188158320405609506988, −8.476610197094911068634084675891, −7.55094784516361353761511570210, −6.61546980019509243437712428500, −5.64448852254817428679515064569, −4.75127434454207242969339712386, −3.70413043769854087468342690817, −2.53687835867811447663255922298, −1.52344328202321822850418770658, 0.73423915449271714187503884576, 2.23091509852329864483018942935, 2.74916042269715485301059886576, 4.49021250771653457460045267612, 5.31547067562833720662951148501, 6.14397470348236770145803749560, 7.14162957565806170358947381981, 8.108942835212444258501246460112, 8.802022057006683479429526313448, 9.462693901985571961969736609145

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