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Properties

Label 2-2793-2793.185-c0-0-0
Degree $2$
Conductor $2793$
Sign $0.233 - 0.972i$
Analytic cond. $1.39388$
Root an. cond. $1.18063$
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  + (0.0249 − 0.999i)3-s + (0.583 + 0.811i)4-s + (−0.124 + 0.992i)7-s + (−0.998 − 0.0498i)9-s + (0.826 − 0.563i)12-s + (−1.57 + 0.662i)13-s + (−0.318 + 0.947i)16-s + (0.222 + 0.974i)19-s + (0.988 + 0.149i)21-s + (−0.542 + 0.840i)25-s + (−0.0747 + 0.997i)27-s + (−0.878 + 0.478i)28-s − 0.822·31-s + (−0.542 − 0.840i)36-s + (1.61 − 0.121i)37-s + ⋯
L(s)  = 1  + (0.0249 − 0.999i)3-s + (0.583 + 0.811i)4-s + (−0.124 + 0.992i)7-s + (−0.998 − 0.0498i)9-s + (0.826 − 0.563i)12-s + (−1.57 + 0.662i)13-s + (−0.318 + 0.947i)16-s + (0.222 + 0.974i)19-s + (0.988 + 0.149i)21-s + (−0.542 + 0.840i)25-s + (−0.0747 + 0.997i)27-s + (−0.878 + 0.478i)28-s − 0.822·31-s + (−0.542 − 0.840i)36-s + (1.61 − 0.121i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.233 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.233 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2793\)    =    \(3 \cdot 7^{2} \cdot 19\)
Sign: $0.233 - 0.972i$
Analytic conductor: \(1.39388\)
Root analytic conductor: \(1.18063\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2793} (185, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2793,\ (\ :0),\ 0.233 - 0.972i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.034467542\)
\(L(\frac12)\) \(\approx\) \(1.034467542\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.0249 + 0.999i)T \)
7 \( 1 + (0.124 - 0.992i)T \)
19 \( 1 + (-0.222 - 0.974i)T \)
good2 \( 1 + (-0.583 - 0.811i)T^{2} \)
5 \( 1 + (0.542 - 0.840i)T^{2} \)
11 \( 1 + (-0.0747 + 0.997i)T^{2} \)
13 \( 1 + (1.57 - 0.662i)T + (0.698 - 0.715i)T^{2} \)
17 \( 1 + (-0.318 - 0.947i)T^{2} \)
23 \( 1 + (-0.980 + 0.198i)T^{2} \)
29 \( 1 + (-0.0249 + 0.999i)T^{2} \)
31 \( 1 + 0.822T + T^{2} \)
37 \( 1 + (-1.61 + 0.121i)T + (0.988 - 0.149i)T^{2} \)
41 \( 1 + (-0.542 + 0.840i)T^{2} \)
43 \( 1 + (-1.95 + 0.395i)T + (0.921 - 0.388i)T^{2} \)
47 \( 1 + (0.698 - 0.715i)T^{2} \)
53 \( 1 + (-0.318 + 0.947i)T^{2} \)
59 \( 1 + (0.124 - 0.992i)T^{2} \)
61 \( 1 + (1.30 - 1.27i)T + (0.0249 - 0.999i)T^{2} \)
67 \( 1 + (0.670 + 0.798i)T + (-0.173 + 0.984i)T^{2} \)
71 \( 1 + (0.878 + 0.478i)T^{2} \)
73 \( 1 + (-0.120 - 0.158i)T + (-0.270 + 0.962i)T^{2} \)
79 \( 1 + (-0.168 - 0.463i)T + (-0.766 + 0.642i)T^{2} \)
83 \( 1 + (0.826 - 0.563i)T^{2} \)
89 \( 1 + (0.583 - 0.811i)T^{2} \)
97 \( 1 + (-1.79 + 0.653i)T + (0.766 - 0.642i)T^{2} \)
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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

−9.098063266831098934096939044882, −8.176682460683378091876123818378, −7.49630702712270607314084254984, −7.15693274626707159299792725478, −6.09138333944593411420246672798, −5.63258215108983014161306039455, −4.38584428458752071621665934394, −3.21184039246656447162758041864, −2.43656229732948460514272421017, −1.80138638100576939772116150391, 0.59931913364484919519524697518, 2.33479565731547478357625014926, 3.05602476662605820196293592981, 4.30018392698516630773058599200, 4.83124006063160143150845831932, 5.67246986156023508697751626543, 6.44301048326637958347738234436, 7.41789616757964152372429080630, 7.86135216381556640079250495341, 9.287650258841981754226888512181

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