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Properties

Label 2-2790-5.4-c1-0-54
Degree $2$
Conductor $2790$
Sign $0.0235 + 0.999i$
Analytic cond. $22.2782$
Root an. cond. $4.71998$
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  − i·2-s − 4-s + (2.23 − 0.0526i)5-s + 2i·7-s + i·8-s + (−0.0526 − 2.23i)10-s + 0.470·11-s − 6.47i·13-s + 2·14-s + 16-s − 7.04i·17-s + 7.04·19-s + (−2.23 + 0.0526i)20-s − 0.470i·22-s + 6.94i·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (0.999 − 0.0235i)5-s + 0.755i·7-s + 0.353i·8-s + (−0.0166 − 0.706i)10-s + 0.141·11-s − 1.79i·13-s + 0.534·14-s + 0.250·16-s − 1.70i·17-s + 1.61·19-s + (−0.499 + 0.0117i)20-s − 0.100i·22-s + 1.44i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0235 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0235 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2790\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 31\)
Sign: $0.0235 + 0.999i$
Analytic conductor: \(22.2782\)
Root analytic conductor: \(4.71998\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2790} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2790,\ (\ :1/2),\ 0.0235 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.106640124\)
\(L(\frac12)\) \(\approx\) \(2.106640124\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 \)
5 \( 1 + (-2.23 + 0.0526i)T \)
31 \( 1 + T \)
good7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 - 0.470T + 11T^{2} \)
13 \( 1 + 6.47iT - 13T^{2} \)
17 \( 1 + 7.04iT - 17T^{2} \)
19 \( 1 - 7.04T + 19T^{2} \)
23 \( 1 - 6.94iT - 23T^{2} \)
29 \( 1 + 6.94T + 29T^{2} \)
37 \( 1 + 1.78iT - 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 0.210iT - 43T^{2} \)
47 \( 1 - 7.04iT - 47T^{2} \)
53 \( 1 + 3.15iT - 53T^{2} \)
59 \( 1 + 7.15T + 59T^{2} \)
61 \( 1 - 11.2T + 61T^{2} \)
67 \( 1 + 8.26iT - 67T^{2} \)
71 \( 1 + 0.260T + 71T^{2} \)
73 \( 1 + 11.8iT - 73T^{2} \)
79 \( 1 - 1.89T + 79T^{2} \)
83 \( 1 + 11.7iT - 83T^{2} \)
89 \( 1 - 12.0T + 89T^{2} \)
97 \( 1 + 3.52iT - 97T^{2} \)
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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

−9.050571595616411472124151095253, −7.81849163980282133141721908362, −7.32447665526712799931346977387, −5.96807425500609589463371179240, −5.37326059016530776534817297812, −5.03222681457935827246980558803, −3.35340912534452124701244545190, −2.94367296023099557841837068034, −1.89776106493156159663114676105, −0.75792939333163638015589463304, 1.21612246439137932815143577702, 2.16351428838903231664165113571, 3.65404957519421756327404047465, 4.30609868242119966551790822875, 5.26765270771681035386536696096, 6.04463519627878470491845642003, 6.76132400941566487411732425576, 7.21289214385532651497717921011, 8.285494449632933689188766654107, 8.964770598895627632474784165002

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