Properties

Label 2-960-12.11-c1-0-25
Degree $2$
Conductor $960$
Sign $0.169 + 0.985i$
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  + (0.292 + 1.70i)3-s i·5-s − 3.41i·7-s + (−2.82 + i)9-s − 2.82·11-s + 2·13-s + (1.70 − 0.292i)15-s − 7.65i·17-s + 2.82i·19-s + (5.82 − i)21-s − 7.41·23-s − 25-s + (−2.53 − 4.53i)27-s − 8i·29-s − 5.65i·31-s + ⋯
L(s)  = 1  + (0.169 + 0.985i)3-s − 0.447i·5-s − 1.29i·7-s + (−0.942 + 0.333i)9-s − 0.852·11-s + 0.554·13-s + (0.440 − 0.0756i)15-s − 1.85i·17-s + 0.648i·19-s + (1.27 − 0.218i)21-s − 1.54·23-s − 0.200·25-s + (−0.487 − 0.872i)27-s − 1.48i·29-s − 1.01i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.169 + 0.985i$
Analytic conductor: \(7.66563\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ 0.169 + 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.823912 - 0.694589i\)
\(L(\frac12)\) \(\approx\) \(0.823912 - 0.694589i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.292 - 1.70i)T \)
5 \( 1 + iT \)
good7 \( 1 + 3.41iT - 7T^{2} \)
11 \( 1 + 2.82T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + 7.65iT - 17T^{2} \)
19 \( 1 - 2.82iT - 19T^{2} \)
23 \( 1 + 7.41T + 23T^{2} \)
29 \( 1 + 8iT - 29T^{2} \)
31 \( 1 + 5.65iT - 31T^{2} \)
37 \( 1 - 0.343T + 37T^{2} \)
41 \( 1 + 2iT - 41T^{2} \)
43 \( 1 - 7.89iT - 43T^{2} \)
47 \( 1 - 6.24T + 47T^{2} \)
53 \( 1 + 3.65iT - 53T^{2} \)
59 \( 1 - 1.65T + 59T^{2} \)
61 \( 1 + 5.65T + 61T^{2} \)
67 \( 1 + 1.07iT - 67T^{2} \)
71 \( 1 + 1.17T + 71T^{2} \)
73 \( 1 - 15.6T + 73T^{2} \)
79 \( 1 + 4.48iT - 79T^{2} \)
83 \( 1 - 5.07T + 83T^{2} \)
89 \( 1 - 7.31iT - 89T^{2} \)
97 \( 1 - 18.9T + 97T^{2} \)
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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

−9.869113963538283544560788599757, −9.263786899534983085947992525334, −7.968442973873206179526497107237, −7.72685636742223541199245844245, −6.26216519437737281059893779057, −5.30510685353685002044739947340, −4.38368658183573462338288319609, −3.74522112318939901511423993579, −2.45284117061914577248896877768, −0.47026160017617460907234407445, 1.73612211265992309580400064701, 2.60689882874429319990856351568, 3.66182829044067056772071800503, 5.33476764644705104083583278791, 6.00001014784392624777439472446, 6.73000673676486891701030068145, 7.82394939994925704118089280682, 8.477999427876743863663141121432, 9.060061852513993294726934255994, 10.41545090147013540728386669210

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