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Properties

Label 2-960-120.29-c2-0-53
Degree $2$
Conductor $960$
Sign $0.999 + 0.0113i$
Analytic cond. $26.1581$
Root an. cond. $5.11449$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.619 + 2.93i)3-s + (4.48 − 2.21i)5-s − 4.63i·7-s + (−8.23 − 3.63i)9-s − 15.0·11-s − 3.35·13-s + (3.71 + 14.5i)15-s + 11.1·17-s + 22.7i·19-s + (13.5 + 2.87i)21-s + 36.1·23-s + (15.2 − 19.8i)25-s + (15.7 − 21.9i)27-s + 21.5·29-s − 28.3·31-s + ⋯
L(s)  = 1  + (−0.206 + 0.978i)3-s + (0.896 − 0.442i)5-s − 0.661i·7-s + (−0.914 − 0.404i)9-s − 1.37·11-s − 0.258·13-s + (0.247 + 0.968i)15-s + 0.655·17-s + 1.19i·19-s + (0.647 + 0.136i)21-s + 1.57·23-s + (0.608 − 0.793i)25-s + (0.584 − 0.811i)27-s + 0.743·29-s − 0.915·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0113i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0113i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.999 + 0.0113i$
Analytic conductor: \(26.1581\)
Root analytic conductor: \(5.11449\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (929, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1),\ 0.999 + 0.0113i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.840534467\)
\(L(\frac12)\) \(\approx\) \(1.840534467\)
\(L(2)\) 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 \( 1 + (0.619 - 2.93i)T \)
5 \( 1 + (-4.48 + 2.21i)T \)
good7 \( 1 + 4.63iT - 49T^{2} \)
11 \( 1 + 15.0T + 121T^{2} \)
13 \( 1 + 3.35T + 169T^{2} \)
17 \( 1 - 11.1T + 289T^{2} \)
19 \( 1 - 22.7iT - 361T^{2} \)
23 \( 1 - 36.1T + 529T^{2} \)
29 \( 1 - 21.5T + 841T^{2} \)
31 \( 1 + 28.3T + 961T^{2} \)
37 \( 1 - 69.5T + 1.36e3T^{2} \)
41 \( 1 + 62.5iT - 1.68e3T^{2} \)
43 \( 1 - 55.2T + 1.84e3T^{2} \)
47 \( 1 + 36.5T + 2.20e3T^{2} \)
53 \( 1 + 29.2iT - 2.80e3T^{2} \)
59 \( 1 - 8.07T + 3.48e3T^{2} \)
61 \( 1 + 10.5iT - 3.72e3T^{2} \)
67 \( 1 - 91.8T + 4.48e3T^{2} \)
71 \( 1 + 79.2iT - 5.04e3T^{2} \)
73 \( 1 + 59.0iT - 5.32e3T^{2} \)
79 \( 1 - 86.3T + 6.24e3T^{2} \)
83 \( 1 + 10.0iT - 6.88e3T^{2} \)
89 \( 1 - 17.8iT - 7.92e3T^{2} \)
97 \( 1 + 139. iT - 9.40e3T^{2} \)
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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

−9.911832783619731734948350035162, −9.238736977190692629536886243797, −8.270864140362970341698251162292, −7.38922614433134979431112222865, −6.09155167845938209923328990823, −5.35909551182369560821035775268, −4.71858359974120459439674474551, −3.54673891025037816673125211454, −2.43511535399902183428853146262, −0.74750682610145112406343622673, 1.00567875856491614186749697290, 2.58730743862092467623139829821, 2.73976409643881691005182224537, 4.99384805360928428291115477795, 5.53572464143111386886219401670, 6.43294932949215220742799050330, 7.24497744958126065379569058200, 8.017660039322850864157908498624, 9.012754634003112014930302450573, 9.771839956220385764674734269127

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