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Properties

Label 2-960-15.8-c1-0-1
Degree $2$
Conductor $960$
Sign $-0.998 - 0.0618i$
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  + (0.292 + 1.70i)3-s + (−0.707 − 2.12i)5-s + (−1 + i)7-s + (−2.82 + i)9-s + 1.41i·11-s + (3.41 − 1.82i)15-s + (−1.41 − 1.41i)17-s + 4i·19-s + (−2 − 1.41i)21-s + (−2.82 + 2.82i)23-s + (−3.99 + 3i)25-s + (−2.53 − 4.53i)27-s − 7.07·29-s − 2·31-s + (−2.41 + 0.414i)33-s + ⋯
L(s)  = 1  + (0.169 + 0.985i)3-s + (−0.316 − 0.948i)5-s + (−0.377 + 0.377i)7-s + (−0.942 + 0.333i)9-s + 0.426i·11-s + (0.881 − 0.472i)15-s + (−0.342 − 0.342i)17-s + 0.917i·19-s + (−0.436 − 0.308i)21-s + (−0.589 + 0.589i)23-s + (−0.799 + 0.600i)25-s + (−0.487 − 0.872i)27-s − 1.31·29-s − 0.359·31-s + (−0.420 + 0.0721i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0618i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-0.998 - 0.0618i$
Analytic conductor: \(7.66563\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (833, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ -0.998 - 0.0618i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0149223 + 0.481948i\)
\(L(\frac12)\) \(\approx\) \(0.0149223 + 0.481948i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.292 - 1.70i)T \)
5 \( 1 + (0.707 + 2.12i)T \)
good7 \( 1 + (1 - i)T - 7iT^{2} \)
11 \( 1 - 1.41iT - 11T^{2} \)
13 \( 1 + 13iT^{2} \)
17 \( 1 + (1.41 + 1.41i)T + 17iT^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + (2.82 - 2.82i)T - 23iT^{2} \)
29 \( 1 + 7.07T + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + (6 - 6i)T - 37iT^{2} \)
41 \( 1 + 5.65iT - 41T^{2} \)
43 \( 1 + (6 + 6i)T + 43iT^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 + (2.82 - 2.82i)T - 53iT^{2} \)
59 \( 1 - 9.89T + 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 + (-4 + 4i)T - 67iT^{2} \)
71 \( 1 - 14.1iT - 71T^{2} \)
73 \( 1 + (5 + 5i)T + 73iT^{2} \)
79 \( 1 + 6iT - 79T^{2} \)
83 \( 1 + (8.48 - 8.48i)T - 83iT^{2} \)
89 \( 1 + 2.82T + 89T^{2} \)
97 \( 1 + (-3 + 3i)T - 97iT^{2} \)
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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

−10.16423732328131222046756948721, −9.609159076809507427011314821273, −8.852066666520139752078336198562, −8.209684779455155106840401339326, −7.18418168116567936861454056584, −5.77971823397591661707713077093, −5.21170507379327805897836170797, −4.16881072783887814163433241579, −3.44324479722191991400280460006, −1.95033313943590074673392815114, 0.20414930739997429319855586129, 2.00955931614056293168803279173, 3.07066979675016612711775305308, 3.94613307147723068915293167730, 5.50880304758178834934059076145, 6.52327839199600418979842332525, 6.95583414335382837895214986669, 7.83053512677344210999360441921, 8.598377603337317095273895214900, 9.599009338647395696682701890989

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