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Properties

Label 1-2793-2793.2081-r0-0-0
Degree $1$
Conductor $2793$
Sign $-0.425 - 0.904i$
Analytic cond. $12.9706$
Root an. cond. $12.9706$
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  + (0.980 − 0.198i)2-s + (0.921 − 0.388i)4-s + (0.583 − 0.811i)5-s + (0.826 − 0.563i)8-s + (0.411 − 0.911i)10-s + (0.988 + 0.149i)11-s + (−0.878 − 0.478i)13-s + (0.698 − 0.715i)16-s + (−0.921 − 0.388i)17-s + (0.222 − 0.974i)20-s + (0.998 − 0.0498i)22-s + (0.124 − 0.992i)23-s + (−0.318 − 0.947i)25-s + (−0.955 − 0.294i)26-s + (0.542 − 0.840i)29-s + ⋯
L(s)  = 1  + (0.980 − 0.198i)2-s + (0.921 − 0.388i)4-s + (0.583 − 0.811i)5-s + (0.826 − 0.563i)8-s + (0.411 − 0.911i)10-s + (0.988 + 0.149i)11-s + (−0.878 − 0.478i)13-s + (0.698 − 0.715i)16-s + (−0.921 − 0.388i)17-s + (0.222 − 0.974i)20-s + (0.998 − 0.0498i)22-s + (0.124 − 0.992i)23-s + (−0.318 − 0.947i)25-s + (−0.955 − 0.294i)26-s + (0.542 − 0.840i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.425 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.425 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2793\)    =    \(3 \cdot 7^{2} \cdot 19\)
Sign: $-0.425 - 0.904i$
Analytic conductor: \(12.9706\)
Root analytic conductor: \(12.9706\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2793} (2081, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2793,\ (0:\ ),\ -0.425 - 0.904i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.835892567 - 2.893083407i\)
\(L(\frac12)\) \(\approx\) \(1.835892567 - 2.893083407i\)
\(L(1)\) \(\approx\) \(1.857134178 - 0.9133383995i\)
\(L(1)\) \(\approx\) \(1.857134178 - 0.9133383995i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.980 - 0.198i)T \)
5 \( 1 + (0.583 - 0.811i)T \)
11 \( 1 + (0.988 + 0.149i)T \)
13 \( 1 + (-0.878 - 0.478i)T \)
17 \( 1 + (-0.921 - 0.388i)T \)
23 \( 1 + (0.124 - 0.992i)T \)
29 \( 1 + (0.542 - 0.840i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.955 + 0.294i)T \)
41 \( 1 + (-0.583 + 0.811i)T \)
43 \( 1 + (-0.969 - 0.246i)T \)
47 \( 1 + (-0.878 - 0.478i)T \)
53 \( 1 + (0.921 - 0.388i)T \)
59 \( 1 + (0.270 + 0.962i)T \)
61 \( 1 + (0.542 - 0.840i)T \)
67 \( 1 + (-0.766 + 0.642i)T \)
71 \( 1 + (0.456 - 0.889i)T \)
73 \( 1 + (-0.0249 + 0.999i)T \)
79 \( 1 + (0.939 - 0.342i)T \)
83 \( 1 + (-0.365 + 0.930i)T \)
89 \( 1 + (0.980 + 0.198i)T \)
97 \( 1 + (0.939 - 0.342i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.63561364580688756869195448019, −18.89070821889699879886683481793, −17.76361271798172106006991322280, −17.31996445183525166322355228734, −16.59604700353587963966930479316, −15.77046469266368128674464718445, −14.86754336318289520149688283806, −14.59523383852351029886646133060, −13.80972564544417913239675731463, −13.28994488455199719832559926999, −12.3521262277883137752645126919, −11.68376756915529601213248223953, −11.02517601284157128833899923198, −10.3145686253933571398755240571, −9.39979618321588026705449922732, −8.62027914812998411350164738207, −7.38705296537989676776452802935, −6.90218341839464790243147455512, −6.34332633468831410344522925425, −5.473150120983403706556959001613, −4.74778828127495643855423412136, −3.73816532093405150005879676270, −3.211121958543969092911697002983, −2.09993487730434020859528594590, −1.65234557725429515011076608076, 0.65476687516510098820056677481, 1.769744832825668068008648823394, 2.3497656634548874144698076864, 3.378825322238102414559522623624, 4.36186702266025603360104324358, 4.871308174308353073244162029185, 5.57310991348800448449884504472, 6.53574877666167801025346271399, 6.95648859051644941487978373503, 8.14264656306366889550322166401, 8.95121322690256904812870636957, 9.82617812107376227968891217919, 10.340814735276561886406774856545, 11.43302725154337797761103827610, 12.00648259294101747029935012758, 12.64538681267623952041014667881, 13.32412595274793556875805888941, 13.90179040933545936510984199789, 14.73853866887183078539914288390, 15.21658726200295318435512703982, 16.23341247759763672290249767458, 16.74773175624691958066656346775, 17.42155984114205248828978406538, 18.23763249836792455525994757432, 19.35185590110061592924151528327

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