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Properties

Label 2-2940-420.299-c0-0-21
Degree $2$
Conductor $2940$
Sign $0.444 - 0.895i$
Analytic cond. $1.46725$
Root an. cond. $1.21130$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)2-s + (0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + (0.499 + 0.866i)6-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (0.866 − 0.499i)10-s + 0.999i·12-s + (0.866 − 0.499i)15-s + (−0.5 + 0.866i)16-s + 0.999i·18-s + 0.999·20-s + (−0.866 − 0.5i)23-s + (−0.5 + 0.866i)24-s + (−0.499 − 0.866i)25-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + (0.499 + 0.866i)6-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (0.866 − 0.499i)10-s + 0.999i·12-s + (0.866 − 0.499i)15-s + (−0.5 + 0.866i)16-s + 0.999i·18-s + 0.999·20-s + (−0.866 − 0.5i)23-s + (−0.5 + 0.866i)24-s + (−0.499 − 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2940\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.444 - 0.895i$
Analytic conductor: \(1.46725\)
Root analytic conductor: \(1.21130\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2940} (1979, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2940,\ (\ :0),\ 0.444 - 0.895i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.941635305\)
\(L(\frac12)\) \(\approx\) \(2.941635305\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 \)
good11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + 1.73iT - T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 + 1.73T + T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + 1.73T + T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−8.695496312592729635398206362020, −8.418552313962216999475687935815, −7.67245337723725177571951991086, −6.72101449830145988167287618551, −5.84905343841636601328963188542, −5.10707896975485862670736829097, −4.37217292180684241515694397461, −3.75575972038887152703012109411, −2.63356709781283595671274250554, −1.85236679601173407319708334497, 1.55745997527236133285536457076, 2.22132576484048982284946758877, 3.27431960975081508271738885242, 3.60689110370965663008295093127, 4.87992972609607544174460302970, 5.75200316547925334434837149483, 6.66962177061105130637044725774, 6.97737451437916065325542458711, 7.954992961259606676796309115334, 8.875560631078001665627907992404

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