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Properties

Label 2-2793-2793.1472-c0-0-0
Degree $2$
Conductor $2793$
Sign $0.999 - 0.00868i$
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  + (−0.318 − 0.947i)3-s + (−0.969 − 0.246i)4-s + (−0.998 + 0.0498i)7-s + (−0.797 + 0.603i)9-s + (0.0747 + 0.997i)12-s + (−0.603 + 1.17i)13-s + (0.878 + 0.478i)16-s + (−0.222 − 0.974i)19-s + (0.365 + 0.930i)21-s + (0.921 + 0.388i)25-s + (0.826 + 0.563i)27-s + (0.980 + 0.198i)28-s + 1.39·31-s + (0.921 − 0.388i)36-s + (−1.60 + 1.09i)37-s + ⋯
L(s)  = 1  + (−0.318 − 0.947i)3-s + (−0.969 − 0.246i)4-s + (−0.998 + 0.0498i)7-s + (−0.797 + 0.603i)9-s + (0.0747 + 0.997i)12-s + (−0.603 + 1.17i)13-s + (0.878 + 0.478i)16-s + (−0.222 − 0.974i)19-s + (0.365 + 0.930i)21-s + (0.921 + 0.388i)25-s + (0.826 + 0.563i)27-s + (0.980 + 0.198i)28-s + 1.39·31-s + (0.921 − 0.388i)36-s + (−1.60 + 1.09i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2793\)    =    \(3 \cdot 7^{2} \cdot 19\)
Sign: $0.999 - 0.00868i$
Analytic conductor: \(1.39388\)
Root analytic conductor: \(1.18063\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2793} (1472, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2793,\ (\ :0),\ 0.999 - 0.00868i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5773851090\)
\(L(\frac12)\) \(\approx\) \(0.5773851090\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.318 + 0.947i)T \)
7 \( 1 + (0.998 - 0.0498i)T \)
19 \( 1 + (0.222 + 0.974i)T \)
good2 \( 1 + (0.969 + 0.246i)T^{2} \)
5 \( 1 + (-0.921 - 0.388i)T^{2} \)
11 \( 1 + (-0.826 - 0.563i)T^{2} \)
13 \( 1 + (0.603 - 1.17i)T + (-0.583 - 0.811i)T^{2} \)
17 \( 1 + (-0.878 + 0.478i)T^{2} \)
23 \( 1 + (0.853 - 0.521i)T^{2} \)
29 \( 1 + (0.318 + 0.947i)T^{2} \)
31 \( 1 - 1.39T + T^{2} \)
37 \( 1 + (1.60 - 1.09i)T + (0.365 - 0.930i)T^{2} \)
41 \( 1 + (-0.921 - 0.388i)T^{2} \)
43 \( 1 + (-1.36 + 0.831i)T + (0.456 - 0.889i)T^{2} \)
47 \( 1 + (0.583 + 0.811i)T^{2} \)
53 \( 1 + (-0.878 - 0.478i)T^{2} \)
59 \( 1 + (0.998 - 0.0498i)T^{2} \)
61 \( 1 + (0.815 - 1.13i)T + (-0.318 - 0.947i)T^{2} \)
67 \( 1 + (-1.24 + 0.452i)T + (0.766 - 0.642i)T^{2} \)
71 \( 1 + (-0.980 + 0.198i)T^{2} \)
73 \( 1 + (-0.293 + 0.455i)T + (-0.411 - 0.911i)T^{2} \)
79 \( 1 + (-0.345 - 1.95i)T + (-0.939 + 0.342i)T^{2} \)
83 \( 1 + (-0.0747 - 0.997i)T^{2} \)
89 \( 1 + (0.969 - 0.246i)T^{2} \)
97 \( 1 + (0.254 + 1.44i)T + (-0.939 + 0.342i)T^{2} \)
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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

−8.910727232323972317516818290221, −8.417494404990972301173773542721, −7.22694415243442312922331502486, −6.77983639450923847517581419097, −6.03483500302449925681683293373, −5.10900883469785557799520486524, −4.45169598391953047771064180093, −3.26615507090109794230970390369, −2.27239968447369710819652815882, −0.908200707154055944671163961136, 0.53755670340749483201835547106, 2.79947811119206093703313204460, 3.46010704498771148495438909861, 4.24437733571727832546668660550, 5.06347052563714553901965333053, 5.73069232977959836613621207126, 6.51626951367571897819637145987, 7.65808354675653694201231440357, 8.402267572006393296632377619629, 9.116017824367670115746089140467

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