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Properties

Label 2-2793-2793.59-c0-0-0
Degree $2$
Conductor $2793$
Sign $0.320 - 0.947i$
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  + (−0.995 − 0.0995i)3-s + (0.797 + 0.603i)4-s + (0.878 + 0.478i)7-s + (0.980 + 0.198i)9-s + (−0.733 − 0.680i)12-s + (0.0291 + 1.16i)13-s + (0.270 + 0.962i)16-s + (−0.623 + 0.781i)19-s + (−0.826 − 0.563i)21-s + (0.661 − 0.749i)25-s + (−0.955 − 0.294i)27-s + (0.411 + 0.911i)28-s − 0.248·31-s + (0.661 + 0.749i)36-s + (−0.355 − 1.15i)37-s + ⋯
L(s)  = 1  + (−0.995 − 0.0995i)3-s + (0.797 + 0.603i)4-s + (0.878 + 0.478i)7-s + (0.980 + 0.198i)9-s + (−0.733 − 0.680i)12-s + (0.0291 + 1.16i)13-s + (0.270 + 0.962i)16-s + (−0.623 + 0.781i)19-s + (−0.826 − 0.563i)21-s + (0.661 − 0.749i)25-s + (−0.955 − 0.294i)27-s + (0.411 + 0.911i)28-s − 0.248·31-s + (0.661 + 0.749i)36-s + (−0.355 − 1.15i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.320 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.320 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2793\)    =    \(3 \cdot 7^{2} \cdot 19\)
Sign: $0.320 - 0.947i$
Analytic conductor: \(1.39388\)
Root analytic conductor: \(1.18063\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2793} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2793,\ (\ :0),\ 0.320 - 0.947i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.168127500\)
\(L(\frac12)\) \(\approx\) \(1.168127500\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.995 + 0.0995i)T \)
7 \( 1 + (-0.878 - 0.478i)T \)
19 \( 1 + (0.623 - 0.781i)T \)
good2 \( 1 + (-0.797 - 0.603i)T^{2} \)
5 \( 1 + (-0.661 + 0.749i)T^{2} \)
11 \( 1 + (-0.955 - 0.294i)T^{2} \)
13 \( 1 + (-0.0291 - 1.16i)T + (-0.998 + 0.0498i)T^{2} \)
17 \( 1 + (0.270 - 0.962i)T^{2} \)
23 \( 1 + (-0.698 + 0.715i)T^{2} \)
29 \( 1 + (0.995 + 0.0995i)T^{2} \)
31 \( 1 + 0.248T + T^{2} \)
37 \( 1 + (0.355 + 1.15i)T + (-0.826 + 0.563i)T^{2} \)
41 \( 1 + (0.661 - 0.749i)T^{2} \)
43 \( 1 + (1.36 - 1.40i)T + (-0.0249 - 0.999i)T^{2} \)
47 \( 1 + (-0.998 + 0.0498i)T^{2} \)
53 \( 1 + (0.270 + 0.962i)T^{2} \)
59 \( 1 + (-0.878 - 0.478i)T^{2} \)
61 \( 1 + (-0.0989 + 1.98i)T + (-0.995 - 0.0995i)T^{2} \)
67 \( 1 + (-0.555 + 1.52i)T + (-0.766 - 0.642i)T^{2} \)
71 \( 1 + (-0.411 + 0.911i)T^{2} \)
73 \( 1 + (0.405 - 0.662i)T + (-0.456 - 0.889i)T^{2} \)
79 \( 1 + (-1.65 - 0.291i)T + (0.939 + 0.342i)T^{2} \)
83 \( 1 + (-0.733 - 0.680i)T^{2} \)
89 \( 1 + (0.797 - 0.603i)T^{2} \)
97 \( 1 + (0.126 - 0.719i)T + (-0.939 - 0.342i)T^{2} \)
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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

−9.052942964092469222530198109160, −8.180479836816819405950066189998, −7.63440060872504004188407868388, −6.60127022981642426696479412788, −6.36173208351702387120660740339, −5.28457978908542921449131386640, −4.52133199842339210066712441502, −3.66798667894789175671536885976, −2.24111002940490541059634356186, −1.60367011562447023104986250902, 0.876152224907721286852884589432, 1.84755600743347367936142450609, 3.11877183712131143467764878037, 4.33644953942454780574853568770, 5.23053078649997077764970330092, 5.54974581311665881689706655915, 6.67198288578743062934858346463, 7.07006309192422271934809489011, 7.905029934748291642530194987633, 8.824791817086813376618559023465

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