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Properties

Label 2-2940-420.299-c0-0-8
Degree $2$
Conductor $2940$
Sign $-0.436 - 0.899i$
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  + (−0.793 + 0.608i)2-s + (0.5 + 0.866i)3-s + (0.258 − 0.965i)4-s + (0.866 + 0.5i)5-s + (−0.923 − 0.382i)6-s + (0.382 + 0.923i)8-s + (−0.499 + 0.866i)9-s + (−0.991 + 0.130i)10-s + (0.965 − 0.258i)12-s + 0.999i·15-s + (−0.866 − 0.499i)16-s + (1.22 − 0.707i)17-s + (−0.130 − 0.991i)18-s + (−0.382 + 0.662i)19-s + (0.707 − 0.707i)20-s + ⋯
L(s)  = 1  + (−0.793 + 0.608i)2-s + (0.5 + 0.866i)3-s + (0.258 − 0.965i)4-s + (0.866 + 0.5i)5-s + (−0.923 − 0.382i)6-s + (0.382 + 0.923i)8-s + (−0.499 + 0.866i)9-s + (−0.991 + 0.130i)10-s + (0.965 − 0.258i)12-s + 0.999i·15-s + (−0.866 − 0.499i)16-s + (1.22 − 0.707i)17-s + (−0.130 − 0.991i)18-s + (−0.382 + 0.662i)19-s + (0.707 − 0.707i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.436 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.436 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2940\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.436 - 0.899i$
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: \((2,\ 2940,\ (\ :0),\ -0.436 - 0.899i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.193231997\)
\(L(\frac12)\) \(\approx\) \(1.193231997\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.793 - 0.608i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 \)
good11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.382 - 0.662i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-1.60 - 0.923i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.923 + 1.60i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.382 + 0.662i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.662 - 0.382i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + 1.41T + T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
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

−9.316059376123937056377246976021, −8.563331405167454681646266997903, −7.64605526951419408778918992101, −7.18346711506256954073494356544, −6.01704686405892333449995681252, −5.52969578420642379172913518044, −4.76700375746865135227578320622, −3.46044079986222595079663220181, −2.60880402423317967372835098413, −1.52598558911227875566863345536, 1.00740262266153015336262331731, 1.78343637486616950191604051074, 2.74852348863549793869266715047, 3.48060444555606254049160881834, 4.77794997607308279755434944400, 5.78715083910646301623700362648, 6.77046220687766162620241363143, 7.19218634138357329348809716994, 8.317986320238564114754856674467, 8.643927274816335851581815595047

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