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Properties

Label 2-2940-35.4-c1-0-39
Degree $2$
Conductor $2940$
Sign $-0.657 + 0.753i$
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 + (−0.133 − 2.23i)5-s + (0.499 + 0.866i)9-s + (2 − 3.46i)11-s − 6i·13-s + (1 − 1.99i)15-s + (1.73 + i)17-s + (−3 − 5.19i)19-s + (−1.73 + i)23-s + (−4.96 + 0.598i)25-s + 0.999i·27-s − 6·29-s + (−1 + 1.73i)31-s + (3.46 − 1.99i)33-s + (−3.46 + 2i)37-s + ⋯
L(s)  = 1  + (0.499 + 0.288i)3-s + (−0.0599 − 0.998i)5-s + (0.166 + 0.288i)9-s + (0.603 − 1.04i)11-s − 1.66i·13-s + (0.258 − 0.516i)15-s + (0.420 + 0.242i)17-s + (−0.688 − 1.19i)19-s + (−0.361 + 0.208i)23-s + (−0.992 + 0.119i)25-s + 0.192i·27-s − 1.11·29-s + (−0.179 + 0.311i)31-s + (0.603 − 0.348i)33-s + (−0.569 + 0.328i)37-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2940\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-0.657 + 0.753i$
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: \((2,\ 2940,\ (\ :1/2),\ -0.657 + 0.753i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.536112666\)
\(L(\frac12)\) \(\approx\) \(1.536112666\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.133 + 2.23i)T \)
7 \( 1 \)
good11 \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 6iT - 13T^{2} \)
17 \( 1 + (-1.73 - i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3 + 5.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.73 - i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.46 - 2i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 8T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 + (3.46 - 2i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.19 - 3i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7 - 12.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.46 + 2i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-8.66 - 5i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 16iT - 83T^{2} \)
89 \( 1 + (4 + 6.92i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 10iT - 97T^{2} \)
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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

−8.564756677825347584841150908457, −7.989704512042678861682552070788, −7.12253429172241047000037841474, −5.94484160721340019806129844978, −5.42494547525023399183097022369, −4.53704001163914126069939408627, −3.62763645680479679820125314504, −2.93797129082118031697706793239, −1.57630842340526330756933027956, −0.42934143909950737326126646051, 1.83928915122763683636838296812, 2.14241762726067597876128037637, 3.69454764984263134861524948730, 3.91217647509236235378160640900, 5.15339370950082611779571595839, 6.37839125986678092874563208938, 6.74412715221628965150616550451, 7.43795556014462181697240216807, 8.170782997945810512377140372359, 9.110129696517681201047275734574

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