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Properties

Label 2-960-20.3-c1-0-23
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

\(L(1)\) \(\approx\) \(0.658576 - 1.35975i\)
\(L(\frac12)\) \(\approx\) \(0.658576 - 1.35975i\)
\(L(\frac{3}{2})\) 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

−9.737917522697013223394374894902, −8.743297894624444563950421848099, −8.342654126614825598365340779543, −6.97343639411346956173944416041, −6.46421190551403747775637638970, −5.51452387560408534470351731422, −4.25404491651832324075368928697, −3.29460835951317370743762618443, −2.03687787399810641839267743308, −0.63163117799087919456653130827, 2.20380821604993025046430492389, 2.71658859229280289360806596297, 4.01608091447451616429943325863, 5.12471207753086104470889426111, 6.16886010125049110266306360604, 6.74356477782793864916315887368, 7.83332814554728487031391329429, 9.023617898965587876722787100319, 9.480642369589895293819561767736, 10.07517254115597778173165636330

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