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Properties

Label 2-960-20.3-c1-0-0
Degree $2$
Conductor $960$
Sign $-0.759 - 0.649i$
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 + (−1.32 − 1.80i)5-s + (−1.86 − 1.86i)7-s − 1.00i·9-s − 0.728i·11-s + (3.12 + 3.12i)13-s + (2.20 + 0.342i)15-s + (1.12 − 1.12i)17-s − 3.73·19-s + 2.64·21-s + (−5.83 + 5.83i)23-s + (−1.51 + 4.76i)25-s + (0.707 + 0.707i)27-s − 2.64i·29-s + 6.01i·31-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (−0.590 − 0.807i)5-s + (−0.705 − 0.705i)7-s − 0.333i·9-s − 0.219i·11-s + (0.866 + 0.866i)13-s + (0.570 + 0.0885i)15-s + (0.272 − 0.272i)17-s − 0.856·19-s + 0.576·21-s + (−1.21 + 1.21i)23-s + (−0.303 + 0.952i)25-s + (0.136 + 0.136i)27-s − 0.490i·29-s + 1.07i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.759 - 0.649i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.759 - 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-0.759 - 0.649i$
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: \((2,\ 960,\ (\ :1/2),\ -0.759 - 0.649i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0850167 + 0.230217i\)
\(L(\frac12)\) \(\approx\) \(0.0850167 + 0.230217i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.32 + 1.80i)T \)
good7 \( 1 + (1.86 + 1.86i)T + 7iT^{2} \)
11 \( 1 + 0.728iT - 11T^{2} \)
13 \( 1 + (-3.12 - 3.12i)T + 13iT^{2} \)
17 \( 1 + (-1.12 + 1.12i)T - 17iT^{2} \)
19 \( 1 + 3.73T + 19T^{2} \)
23 \( 1 + (5.83 - 5.83i)T - 23iT^{2} \)
29 \( 1 + 2.64iT - 29T^{2} \)
31 \( 1 - 6.01iT - 31T^{2} \)
37 \( 1 + (3.12 - 3.12i)T - 37iT^{2} \)
41 \( 1 + 4.24T + 41T^{2} \)
43 \( 1 + (5.10 - 5.10i)T - 43iT^{2} \)
47 \( 1 + (-2.09 - 2.09i)T + 47iT^{2} \)
53 \( 1 + (0.484 + 0.484i)T + 53iT^{2} \)
59 \( 1 + 4.92T + 59T^{2} \)
61 \( 1 + 2.31T + 61T^{2} \)
67 \( 1 + (-5.10 - 5.10i)T + 67iT^{2} \)
71 \( 1 + 13.1iT - 71T^{2} \)
73 \( 1 + (-3.96 - 3.96i)T + 73iT^{2} \)
79 \( 1 + 7.11T + 79T^{2} \)
83 \( 1 + (3.55 - 3.55i)T - 83iT^{2} \)
89 \( 1 - 1.03iT - 89T^{2} \)
97 \( 1 + (12.5 - 12.5i)T - 97iT^{2} \)
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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

−10.29002075888769422016334600711, −9.565046086233005601723080466251, −8.739024944121253657182008003269, −7.936774322471497663800292126429, −6.85771137674075765415802529733, −6.08924033642683825515598697384, −5.00281169544195314903824772743, −4.04769400372267765316166839940, −3.47209397191805189820258474514, −1.42513104077381806684918332861, 0.12279315669825173467893959332, 2.15485654641647593802285285238, 3.23420022903287027302721239159, 4.21976788992275237497844836108, 5.68404156834591751434135931546, 6.26110214563913541011664129692, 7.02422785953717677176373198072, 8.080716382383358506956144127832, 8.630732871674755703889171112369, 9.956378910150651762628266966603

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