Properties

Label 2-2940-35.4-c1-0-27
Degree $2$
Conductor $2940$
Sign $0.753 + 0.657i$
Analytic cond. $23.4760$
Root an. cond. $4.84520$
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  + (−0.866 − 0.5i)3-s + (2.23 − 0.133i)5-s + (0.499 + 0.866i)9-s + (2 − 3.46i)11-s + (−1.99 − i)15-s + (3.46 + 2i)17-s + (−3.46 + 2i)23-s + (4.96 − 0.598i)25-s − 0.999i·27-s + 6·29-s + (2 − 3.46i)31-s + (−3.46 + 1.99i)33-s + (−6.92 + 4i)37-s + 10·41-s − 4i·43-s + ⋯
L(s)  = 1  + (−0.499 − 0.288i)3-s + (0.998 − 0.0599i)5-s + (0.166 + 0.288i)9-s + (0.603 − 1.04i)11-s + (−0.516 − 0.258i)15-s + (0.840 + 0.485i)17-s + (−0.722 + 0.417i)23-s + (0.992 − 0.119i)25-s − 0.192i·27-s + 1.11·29-s + (0.359 − 0.622i)31-s + (−0.603 + 0.348i)33-s + (−1.13 + 0.657i)37-s + 1.56·41-s − 0.609i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.753 + 0.657i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.753 + 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2940\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.753 + 0.657i$
Analytic conductor: \(23.4760\)
Root analytic conductor: \(4.84520\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2940} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2940,\ (\ :1/2),\ 0.753 + 0.657i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.034904090\)
\(L(\frac12)\) \(\approx\) \(2.034904090\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (-2.23 + 0.133i)T \)
7 \( 1 \)
good11 \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + (-3.46 - 2i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.46 - 2i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (6.92 - 4i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 + (-3.46 + 2i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (10.3 + 6i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.46 - 2i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-6.92 - 4i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6 + 10.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
89 \( 1 + (-5 - 8.66i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 8iT - 97T^{2} \)
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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

−8.624807293544545280573865391233, −8.000312433135277015533915999809, −6.97676365968427584108296907095, −6.17691336383326233273012993272, −5.82680832856455632703707059416, −4.99568202989921654313559139755, −3.90947180048060406283874341320, −2.91588936188680976752005186297, −1.77386860575861602152715686459, −0.845782961258970019901223728095, 1.07331242778245243522132327064, 2.11794387766076669046191034499, 3.16339255585273611147973998494, 4.34758824835669358294309739775, 4.95794196273389300800434867635, 5.82845157108284219333384889279, 6.46982464763614533738422632785, 7.15769168720459446717107125232, 8.071694845612806022153552016698, 9.148473332441876864083001953283

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