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Properties

Label 1-828-828.655-r0-0-0
Degree $1$
Conductor $828$
Sign $0.268 - 0.963i$
Analytic cond. $3.84521$
Root an. cond. $3.84521$
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  + (−0.0475 − 0.998i)5-s + (0.928 − 0.371i)7-s + (0.580 − 0.814i)11-s + (0.928 + 0.371i)13-s + (0.654 − 0.755i)17-s + (−0.654 − 0.755i)19-s + (−0.995 + 0.0950i)25-s + (0.981 + 0.189i)29-s + (−0.235 + 0.971i)31-s + (−0.415 − 0.909i)35-s + (−0.841 − 0.540i)37-s + (0.0475 + 0.998i)41-s + (0.235 + 0.971i)43-s + (0.5 + 0.866i)47-s + (0.723 − 0.690i)49-s + ⋯
L(s)  = 1  + (−0.0475 − 0.998i)5-s + (0.928 − 0.371i)7-s + (0.580 − 0.814i)11-s + (0.928 + 0.371i)13-s + (0.654 − 0.755i)17-s + (−0.654 − 0.755i)19-s + (−0.995 + 0.0950i)25-s + (0.981 + 0.189i)29-s + (−0.235 + 0.971i)31-s + (−0.415 − 0.909i)35-s + (−0.841 − 0.540i)37-s + (0.0475 + 0.998i)41-s + (0.235 + 0.971i)43-s + (0.5 + 0.866i)47-s + (0.723 − 0.690i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.268 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.268 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(828\)    =    \(2^{2} \cdot 3^{2} \cdot 23\)
Sign: $0.268 - 0.963i$
Analytic conductor: \(3.84521\)
Root analytic conductor: \(3.84521\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{828} (655, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 828,\ (0:\ ),\ 0.268 - 0.963i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.378962219 - 1.047190990i\)
\(L(\frac12)\) \(\approx\) \(1.378962219 - 1.047190990i\)
\(L(1)\) \(\approx\) \(1.195051511 - 0.3913577728i\)
\(L(1)\) \(\approx\) \(1.195051511 - 0.3913577728i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
23 \( 1 \)
good5 \( 1 + (-0.0475 - 0.998i)T \)
7 \( 1 + (0.928 - 0.371i)T \)
11 \( 1 + (0.580 - 0.814i)T \)
13 \( 1 + (0.928 + 0.371i)T \)
17 \( 1 + (0.654 - 0.755i)T \)
19 \( 1 + (-0.654 - 0.755i)T \)
29 \( 1 + (0.981 + 0.189i)T \)
31 \( 1 + (-0.235 + 0.971i)T \)
37 \( 1 + (-0.841 - 0.540i)T \)
41 \( 1 + (0.0475 + 0.998i)T \)
43 \( 1 + (0.235 + 0.971i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (0.142 - 0.989i)T \)
59 \( 1 + (-0.928 - 0.371i)T \)
61 \( 1 + (-0.723 - 0.690i)T \)
67 \( 1 + (0.580 + 0.814i)T \)
71 \( 1 + (-0.415 + 0.909i)T \)
73 \( 1 + (-0.654 - 0.755i)T \)
79 \( 1 + (-0.786 + 0.618i)T \)
83 \( 1 + (0.0475 - 0.998i)T \)
89 \( 1 + (0.959 + 0.281i)T \)
97 \( 1 + (0.888 + 0.458i)T \)
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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

−22.39185506576414201251578235044, −21.49351283853762667690799647314, −20.863012243800642120756463204006, −19.979455829111918102862194290629, −18.87869510490459556890485094147, −18.487453843361754063089120809645, −17.52263521357453907058159481858, −16.993531882057677601606221141142, −15.56353589955974313486117205634, −15.076893278059603139871392889929, −14.36507611484456664305321099253, −13.62014551416123002414525899306, −12.308458157057231346618006598586, −11.80809761021001074652151900107, −10.646687136909516591883395160427, −10.332609885904033577130507678893, −9.009326616303753444035760257554, −8.13838341290044995683670693267, −7.37097983369523593045449588550, −6.30928920759719167187635398854, −5.62504714781052257679504967025, −4.30094060116602170649783701787, −3.5555924979407356050395720187, −2.29102181694527629317411074247, −1.46020891424436559620241658551, 0.897711997361304841822775248925, 1.60601503208319836635772983615, 3.14455661620745507881911368815, 4.24620112831777796402326475551, 4.91362083680669716024081643970, 5.90219227168262828197474448947, 6.9346249679645629358041547106, 8.08861386696940419252985666225, 8.66008716301308376125248520567, 9.39804131654746344381635081328, 10.68677736049846266427981682190, 11.38834334750854123225248618122, 12.121282374246342418190514415094, 13.14811960482580203487964435952, 13.938349459443532103618870085711, 14.5094520215220272134694753414, 15.87002371123192701842684745652, 16.29255019806279041420357182708, 17.22678599809975128580421668200, 17.83485989248141581479548838866, 18.8837668608508138601434178850, 19.71993874890387037564200834044, 20.46314863341285777593971906570, 21.28721695047951261065650203198, 21.61650257506521845095884239049

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