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Properties

Label 2-960-15.2-c1-0-16
Degree $2$
Conductor $960$
Sign $0.794 + 0.607i$
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  + (−0.618 − 1.61i)3-s − 2.23·5-s + (1 + i)7-s + (−2.23 + 2.00i)9-s + 4.47i·11-s + (3 − 3i)13-s + (1.38 + 3.61i)15-s + (2.23 − 2.23i)17-s − 2i·19-s + (1 − 2.23i)21-s + (2.23 + 2.23i)23-s + 5.00·25-s + (4.61 + 2.38i)27-s + 4.47·29-s − 4·31-s + ⋯
L(s)  = 1  + (−0.356 − 0.934i)3-s − 0.999·5-s + (0.377 + 0.377i)7-s + (−0.745 + 0.666i)9-s + 1.34i·11-s + (0.832 − 0.832i)13-s + (0.356 + 0.934i)15-s + (0.542 − 0.542i)17-s − 0.458i·19-s + (0.218 − 0.487i)21-s + (0.466 + 0.466i)23-s + 1.00·25-s + (0.888 + 0.458i)27-s + 0.830·29-s − 0.718·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.607i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.794 + 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.794 + 0.607i$
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: \((2,\ 960,\ (\ :1/2),\ 0.794 + 0.607i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13132 - 0.382683i\)
\(L(\frac12)\) \(\approx\) \(1.13132 - 0.382683i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.618 + 1.61i)T \)
5 \( 1 + 2.23T \)
good7 \( 1 + (-1 - i)T + 7iT^{2} \)
11 \( 1 - 4.47iT - 11T^{2} \)
13 \( 1 + (-3 + 3i)T - 13iT^{2} \)
17 \( 1 + (-2.23 + 2.23i)T - 17iT^{2} \)
19 \( 1 + 2iT - 19T^{2} \)
23 \( 1 + (-2.23 - 2.23i)T + 23iT^{2} \)
29 \( 1 - 4.47T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (-3 - 3i)T + 37iT^{2} \)
41 \( 1 + 8.94iT - 41T^{2} \)
43 \( 1 + (3 - 3i)T - 43iT^{2} \)
47 \( 1 + (-6.70 + 6.70i)T - 47iT^{2} \)
53 \( 1 + (2.23 + 2.23i)T + 53iT^{2} \)
59 \( 1 - 8.94T + 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 + (1 + i)T + 67iT^{2} \)
71 \( 1 - 4.47iT - 71T^{2} \)
73 \( 1 + (-1 + i)T - 73iT^{2} \)
79 \( 1 - 6iT - 79T^{2} \)
83 \( 1 + (-6.70 - 6.70i)T + 83iT^{2} \)
89 \( 1 - 4.47T + 89T^{2} \)
97 \( 1 + (-9 - 9i)T + 97iT^{2} \)
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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

−10.06791014122932207296323816876, −8.814531981063534522033881114149, −8.145883112575406936710773497161, −7.34735026262172222945230995298, −6.82317634587274206692428933430, −5.52676295260787150155692616980, −4.84762009877827991294216664479, −3.52262263485401024909124062349, −2.29745741357158006415147540084, −0.873829085063278576179796814230, 0.923981326338813527966777156661, 3.15666881111249739896404793993, 3.89115832870612767749849336164, 4.62584680719924577043834850740, 5.77807111727761727633209688105, 6.54322631091668390311795207544, 7.81227529404294124252224295436, 8.524777819989857651882552003306, 9.150697420989485248023431708584, 10.38608120458597779827990454175

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