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Properties

Label 2-960-240.77-c1-0-20
Degree $2$
Conductor $960$
Sign $0.978 + 0.207i$
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  + (−1.59 − 0.672i)3-s + (1.06 + 1.96i)5-s + (3.58 − 3.58i)7-s + (2.09 + 2.14i)9-s + (1.75 + 1.75i)11-s − 1.86·13-s + (−0.373 − 3.85i)15-s + (2.63 − 2.63i)17-s + (−0.850 + 0.850i)19-s + (−8.14 + 3.31i)21-s + (−2.34 + 2.34i)23-s + (−2.73 + 4.18i)25-s + (−1.89 − 4.83i)27-s + (2.25 + 2.25i)29-s − 3.07·31-s + ⋯
L(s)  = 1  + (−0.921 − 0.388i)3-s + (0.475 + 0.879i)5-s + (1.35 − 1.35i)7-s + (0.698 + 0.715i)9-s + (0.528 + 0.528i)11-s − 0.516·13-s + (−0.0965 − 0.995i)15-s + (0.637 − 0.637i)17-s + (−0.195 + 0.195i)19-s + (−1.77 + 0.722i)21-s + (−0.489 + 0.489i)23-s + (−0.547 + 0.836i)25-s + (−0.365 − 0.930i)27-s + (0.419 + 0.419i)29-s − 0.552·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.207i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 + 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.978 + 0.207i$
Analytic conductor: \(7.66563\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ 0.978 + 0.207i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53231 - 0.160453i\)
\(L(\frac12)\) \(\approx\) \(1.53231 - 0.160453i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.59 + 0.672i)T \)
5 \( 1 + (-1.06 - 1.96i)T \)
good7 \( 1 + (-3.58 + 3.58i)T - 7iT^{2} \)
11 \( 1 + (-1.75 - 1.75i)T + 11iT^{2} \)
13 \( 1 + 1.86T + 13T^{2} \)
17 \( 1 + (-2.63 + 2.63i)T - 17iT^{2} \)
19 \( 1 + (0.850 - 0.850i)T - 19iT^{2} \)
23 \( 1 + (2.34 - 2.34i)T - 23iT^{2} \)
29 \( 1 + (-2.25 - 2.25i)T + 29iT^{2} \)
31 \( 1 + 3.07T + 31T^{2} \)
37 \( 1 - 9.19T + 37T^{2} \)
41 \( 1 - 7.89T + 41T^{2} \)
43 \( 1 + 1.44iT - 43T^{2} \)
47 \( 1 + (-3.16 + 3.16i)T - 47iT^{2} \)
53 \( 1 - 7.55T + 53T^{2} \)
59 \( 1 + (3.61 - 3.61i)T - 59iT^{2} \)
61 \( 1 + (2.53 + 2.53i)T + 61iT^{2} \)
67 \( 1 + 1.34iT - 67T^{2} \)
71 \( 1 - 11.4T + 71T^{2} \)
73 \( 1 + (4.59 + 4.59i)T + 73iT^{2} \)
79 \( 1 + 1.73iT - 79T^{2} \)
83 \( 1 + 0.755iT - 83T^{2} \)
89 \( 1 - 8.15iT - 89T^{2} \)
97 \( 1 + (-1.60 - 1.60i)T + 97iT^{2} \)
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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

−10.26124983247964310525386875385, −9.472272095071619261533272797644, −7.82263400186489921245867259366, −7.43162510995830518911470396065, −6.74155807248287321040948439215, −5.69270433052288160385331848330, −4.78361201716325606951091116022, −3.91647987509228989958623992493, −2.18147432071559677266604558552, −1.10304450100813905586300294853, 1.12689879019977443811469855097, 2.32919320628014223257221921945, 4.19356734482585569396198717170, 4.88583090280118818206441073820, 5.76986437598780753100352740661, 6.10788225244669642965064884596, 7.70645701577406489277214639903, 8.545746221681311366154997489156, 9.207197903053159559987229145985, 9.999554214415126188297925969209

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