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Properties

Label 2-960-15.2-c1-0-38
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: $\chi_{960} (257, \cdot )$
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

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

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