Properties

Label 2-960-20.3-c1-0-12
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

\(L(1)\) \(\approx\) \(1.76891 - 0.653239i\)
\(L(\frac12)\) \(\approx\) \(1.76891 - 0.653239i\)
\(L(\frac{3}{2})\) 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

−9.650774088222718236946518498825, −8.855563579581167267142844802779, −8.414403120470782406223396068756, −7.56340589762472689471903930303, −6.64741899382456178285736345779, −5.46966643193608322486361991562, −4.65333373936817647315363280196, −3.62795440887873634509493742734, −2.27320277860362826138128046133, −1.06796706465958162340596255383, 1.28668591331309933325461287384, 3.13022714980921845150388832238, 3.55556808965196796818498069446, 4.74872201429807708008071960061, 5.71723024155842567499205775521, 6.99206176231691989573696011203, 7.65627021898451910354257669435, 8.316620772046030019705400697634, 9.276744018515309910974220348583, 10.34535893077547394826138931154

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