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Properties

Label 2-2940-420.419-c0-0-17
Degree $2$
Conductor $2940$
Sign $0.981 + 0.188i$
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  + i·2-s + (0.866 − 0.5i)3-s − 4-s − 5-s + (0.5 + 0.866i)6-s − i·8-s + (0.499 − 0.866i)9-s − i·10-s + (−0.866 + 0.5i)12-s + (−0.866 + 0.5i)15-s + 16-s + (0.866 + 0.499i)18-s + 20-s − i·23-s + (−0.5 − 0.866i)24-s + 25-s + ⋯
L(s)  = 1  + i·2-s + (0.866 − 0.5i)3-s − 4-s − 5-s + (0.5 + 0.866i)6-s − i·8-s + (0.499 − 0.866i)9-s − i·10-s + (−0.866 + 0.5i)12-s + (−0.866 + 0.5i)15-s + 16-s + (0.866 + 0.499i)18-s + 20-s − i·23-s + (−0.5 − 0.866i)24-s + 25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2940\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.981 + 0.188i$
Analytic conductor: \(1.46725\)
Root analytic conductor: \(1.21130\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2940} (2939, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2940,\ (\ :0),\ 0.981 + 0.188i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.166559022\)
\(L(\frac12)\) \(\approx\) \(1.166559022\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−8.608795264904144645485582901747, −8.011560546276154861912813544566, −7.60083521367494481202898000200, −6.74448794494007092530463453960, −6.21511784238440030845334729549, −4.98921568910743627821098000625, −4.15165194083351797669435110357, −3.55341432879724087061167590416, −2.41209299527945286093835537457, −0.73738088625517340643680494434, 1.36824532642635361137799105443, 2.58204967386421177430400888226, 3.40048269177227092426632527519, 3.93093551697515629863448623807, 4.76605611428809767672604443837, 5.48555630824489089686621082220, 7.03140100145216526966953526788, 7.71006759957523102522211795369, 8.447073111619063257827158125492, 8.989399579449989905570730291082

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