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Properties

Label 2-960-40.29-c1-0-20
Degree $2$
Conductor $960$
Sign $0.0560 + 0.998i$
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  + 3-s + (−0.456 + 2.18i)5-s − 4.37i·7-s + 9-s − 5.58i·11-s − 4.37·13-s + (−0.456 + 2.18i)15-s − 5.58i·17-s + 4i·19-s − 4.37i·21-s + (−4.58 − 1.99i)25-s + 27-s − 2.55i·29-s − 5.29·31-s − 5.58i·33-s + ⋯
L(s)  = 1  + 0.577·3-s + (−0.204 + 0.978i)5-s − 1.65i·7-s + 0.333·9-s − 1.68i·11-s − 1.21·13-s + (−0.117 + 0.565i)15-s − 1.35i·17-s + 0.917i·19-s − 0.955i·21-s + (−0.916 − 0.399i)25-s + 0.192·27-s − 0.473i·29-s − 0.950·31-s − 0.971i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0560 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0560 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.0560 + 0.998i$
Analytic conductor: \(7.66563\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ 0.0560 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05926 - 1.00150i\)
\(L(\frac12)\) \(\approx\) \(1.05926 - 1.00150i\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 + (0.456 - 2.18i)T \)
good7 \( 1 + 4.37iT - 7T^{2} \)
11 \( 1 + 5.58iT - 11T^{2} \)
13 \( 1 + 4.37T + 13T^{2} \)
17 \( 1 + 5.58iT - 17T^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 2.55iT - 29T^{2} \)
31 \( 1 + 5.29T + 31T^{2} \)
37 \( 1 + 2.55T + 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 11.1T + 43T^{2} \)
47 \( 1 + 6.92iT - 47T^{2} \)
53 \( 1 - 7.84T + 53T^{2} \)
59 \( 1 + 1.58iT - 59T^{2} \)
61 \( 1 - 10.5iT - 61T^{2} \)
67 \( 1 - 3.16T + 67T^{2} \)
71 \( 1 - 6.92T + 71T^{2} \)
73 \( 1 + 12iT - 73T^{2} \)
79 \( 1 - 5.29T + 79T^{2} \)
83 \( 1 + 7.16T + 83T^{2} \)
89 \( 1 + 2T + 89T^{2} \)
97 \( 1 - 11.1iT - 97T^{2} \)
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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

−9.965266253891766129945271765989, −9.062585850330500893700804775788, −7.80862500764860151393831061846, −7.46621453112574973703351386091, −6.68566910118218365546278963836, −5.51954612416174047459302880557, −4.14252665933310965606549318476, −3.46831383338552390302390837986, −2.51053011147161541989957919108, −0.61288261987233158384582365943, 1.85290777608434368963523476933, 2.55132825045072329110727062228, 4.13604111549093705642502881919, 4.94020345652817682347598384140, 5.71298505598631503909741211401, 7.03189858598462201853236789256, 7.83624873746902288866709188310, 8.720176738424043637634655264689, 9.310086737308962918725870432729, 9.820872458787831795032459313760

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