Properties

Label 2-960-15.2-c1-0-8
Degree $2$
Conductor $960$
Sign $-0.391 - 0.920i$
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  + (1.41 + i)3-s + (1 + 2i)5-s + (−2.41 − 2.41i)7-s + (1.00 + 2.82i)9-s + 0.828i·11-s + (−3.82 + 3.82i)13-s + (−0.585 + 3.82i)15-s + (−1.82 + 1.82i)17-s + 0.828i·19-s + (−1 − 5.82i)21-s + (4.41 + 4.41i)23-s + (−3 + 4i)25-s + (−1.41 + 5.00i)27-s − 3.65·29-s + 5.65·31-s + ⋯
L(s)  = 1  + (0.816 + 0.577i)3-s + (0.447 + 0.894i)5-s + (−0.912 − 0.912i)7-s + (0.333 + 0.942i)9-s + 0.249i·11-s + (−1.06 + 1.06i)13-s + (−0.151 + 0.988i)15-s + (−0.443 + 0.443i)17-s + 0.190i·19-s + (−0.218 − 1.27i)21-s + (0.920 + 0.920i)23-s + (−0.600 + 0.800i)25-s + (−0.272 + 0.962i)27-s − 0.679·29-s + 1.01·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.391 - 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.391 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-0.391 - 0.920i$
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: \((2,\ 960,\ (\ :1/2),\ -0.391 - 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.953898 + 1.44167i\)
\(L(\frac12)\) \(\approx\) \(0.953898 + 1.44167i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.41 - i)T \)
5 \( 1 + (-1 - 2i)T \)
good7 \( 1 + (2.41 + 2.41i)T + 7iT^{2} \)
11 \( 1 - 0.828iT - 11T^{2} \)
13 \( 1 + (3.82 - 3.82i)T - 13iT^{2} \)
17 \( 1 + (1.82 - 1.82i)T - 17iT^{2} \)
19 \( 1 - 0.828iT - 19T^{2} \)
23 \( 1 + (-4.41 - 4.41i)T + 23iT^{2} \)
29 \( 1 + 3.65T + 29T^{2} \)
31 \( 1 - 5.65T + 31T^{2} \)
37 \( 1 + (-5.82 - 5.82i)T + 37iT^{2} \)
41 \( 1 + 5.65iT - 41T^{2} \)
43 \( 1 + (-0.414 + 0.414i)T - 43iT^{2} \)
47 \( 1 + (-3.58 + 3.58i)T - 47iT^{2} \)
53 \( 1 + (-3 - 3i)T + 53iT^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + 0.343T + 61T^{2} \)
67 \( 1 + (10.0 + 10.0i)T + 67iT^{2} \)
71 \( 1 + 10.4iT - 71T^{2} \)
73 \( 1 + (4.65 - 4.65i)T - 73iT^{2} \)
79 \( 1 - 0.828iT - 79T^{2} \)
83 \( 1 + (3.24 + 3.24i)T + 83iT^{2} \)
89 \( 1 - 15.6T + 89T^{2} \)
97 \( 1 + (-1 - i)T + 97iT^{2} \)
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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

−10.08663177337134710962980807175, −9.628284028189096304269523018079, −8.900320726419559851593044058006, −7.50245077485152978717561257559, −7.12058179794142324784850000407, −6.17909780067504092181828426320, −4.79947269871103819959803145627, −3.87957185834197958820407617052, −3.02067323182772151912946744306, −1.98675230880547947701805584237, 0.70004685475156905462646791236, 2.41303129375618878177603854585, 2.93007465001577633464200560874, 4.45202681728108806967077049010, 5.53086031412626408072374250709, 6.31993389733964466897643732367, 7.32030653044692250602294183144, 8.248614287583453626071513625288, 8.999664632830784609165068877806, 9.477982681423281168172813199748

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