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Properties

Label 2-2940-35.9-c1-0-24
Degree $2$
Conductor $2940$
Sign $0.997 + 0.0667i$
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 + (1.86 − 1.23i)5-s + (0.499 − 0.866i)9-s + (2 + 3.46i)11-s + 6i·13-s + (1 − 2i)15-s + (1.73 − i)17-s + (3 − 5.19i)19-s + (1.73 + i)23-s + (1.96 − 4.59i)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 + (0.834 − 0.550i)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 + (0.392 − 0.919i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0667i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2940\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.997 + 0.0667i$
Analytic conductor: \(23.4760\)
Root analytic conductor: \(4.84520\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2940} (1549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2940,\ (\ :1/2),\ 0.997 + 0.0667i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.862527760\)
\(L(\frac12)\) \(\approx\) \(2.862527760\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.86 + 1.23i)T \)
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

−9.134553694041127962487654051493, −7.988535059047088270333972348195, −7.09408612694830283936304866673, −6.69151963530290137674039120406, −5.70188717355671023038426728772, −4.72515909994994945672842145294, −4.19025614488811868146398182118, −2.88034595254221481010795903970, −1.95637795947263873429893611643, −1.20603940490055246637402810889, 0.999529305838686521582577084870, 2.21854360894033444636621109786, 3.31976402014669517966946932384, 3.56523953075540899365414786973, 5.05321881105947221911586007869, 5.83025544126487308994576808427, 6.21646530441243172480479871987, 7.49028720208500840332757692192, 7.952692218988161580137076022173, 8.806329634296815589220392986003

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