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Properties

Label 1-828-828.535-r1-0-0
Degree $1$
Conductor $828$
Sign $-0.380 + 0.924i$
Analytic cond. $88.9809$
Root an. cond. $88.9809$
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 + 0.0950i)5-s + (−0.723 + 0.690i)7-s + (0.327 + 0.945i)11-s + (0.723 + 0.690i)13-s + (−0.142 − 0.989i)17-s + (0.142 − 0.989i)19-s + (0.981 − 0.189i)25-s + (0.928 + 0.371i)29-s + (0.888 + 0.458i)31-s + (0.654 − 0.755i)35-s + (0.415 + 0.909i)37-s + (−0.995 + 0.0950i)41-s + (0.888 − 0.458i)43-s + (0.5 − 0.866i)47-s + (0.0475 − 0.998i)49-s + ⋯
L(s)  = 1  + (−0.995 + 0.0950i)5-s + (−0.723 + 0.690i)7-s + (0.327 + 0.945i)11-s + (0.723 + 0.690i)13-s + (−0.142 − 0.989i)17-s + (0.142 − 0.989i)19-s + (0.981 − 0.189i)25-s + (0.928 + 0.371i)29-s + (0.888 + 0.458i)31-s + (0.654 − 0.755i)35-s + (0.415 + 0.909i)37-s + (−0.995 + 0.0950i)41-s + (0.888 − 0.458i)43-s + (0.5 − 0.866i)47-s + (0.0475 − 0.998i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.380 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.380 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(828\)    =    \(2^{2} \cdot 3^{2} \cdot 23\)
Sign: $-0.380 + 0.924i$
Analytic conductor: \(88.9809\)
Root analytic conductor: \(88.9809\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{828} (535, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 828,\ (1:\ ),\ -0.380 + 0.924i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6700647597 + 0.9998901715i\)
\(L(\frac12)\) \(\approx\) \(0.6700647597 + 0.9998901715i\)
\(L(1)\) \(\approx\) \(0.8442559286 + 0.1998166533i\)
\(L(1)\) \(\approx\) \(0.8442559286 + 0.1998166533i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
23 \( 1 \)
good5 \( 1 + (-0.995 + 0.0950i)T \)
7 \( 1 + (-0.723 + 0.690i)T \)
11 \( 1 + (0.327 + 0.945i)T \)
13 \( 1 + (0.723 + 0.690i)T \)
17 \( 1 + (-0.142 - 0.989i)T \)
19 \( 1 + (0.142 - 0.989i)T \)
29 \( 1 + (0.928 + 0.371i)T \)
31 \( 1 + (0.888 + 0.458i)T \)
37 \( 1 + (0.415 + 0.909i)T \)
41 \( 1 + (-0.995 + 0.0950i)T \)
43 \( 1 + (0.888 - 0.458i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (-0.959 - 0.281i)T \)
59 \( 1 + (-0.723 - 0.690i)T \)
61 \( 1 + (0.0475 + 0.998i)T \)
67 \( 1 + (0.327 - 0.945i)T \)
71 \( 1 + (0.654 + 0.755i)T \)
73 \( 1 + (-0.142 + 0.989i)T \)
79 \( 1 + (-0.235 + 0.971i)T \)
83 \( 1 + (0.995 + 0.0950i)T \)
89 \( 1 + (0.841 + 0.540i)T \)
97 \( 1 + (0.580 + 0.814i)T \)
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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

−21.79909477603036910149554664094, −20.758809559820808792196051801038, −20.08512465403680853259254951195, −19.21975891413060145047706328818, −18.90621348310806669120645020394, −17.624255102367378928322100661271, −16.76087529473068928502400755139, −16.06282059986765924789670168708, −15.48046947995269385150614431568, −14.41183218410492074747720810537, −13.56445949480948795005729850128, −12.735324615090660211910722032271, −11.971202786246628894201267214022, −10.910703044139747584443631710856, −10.437967300144396482040098314483, −9.23490509236853941017355773943, −8.2100471595511558904920590668, −7.75467821293627819785976062038, −6.463359620697430625963129310056, −5.90596821648198592537827743984, −4.40178332937510304968254902855, −3.68707503521652240505251870022, −3.016643580639007353111297741619, −1.21483380952158326502869186163, −0.36232636236801815023618351061, 0.94079124352606402559439956717, 2.428525671900493024649981725406, 3.28650184181608577976188153508, 4.34478027141105902303477419865, 5.11691019694750400175027236392, 6.657434766019303195773941733269, 6.86917005343661529643431032748, 8.16460414761874921488681063008, 8.99423173195583865208516046628, 9.70598661836567356126256149610, 10.85738858085777063937283992503, 11.82995189526782450166949239300, 12.1645292501876919675180505344, 13.25528780509410588329244316470, 14.13563710008473014501848904468, 15.25870162105526432324714425785, 15.68692112492968043417547042742, 16.36227466797876716609912376900, 17.46978616140351777143150394918, 18.43010382399475506443157654381, 18.981995142146862740647346185318, 19.87432834056231829835275314380, 20.397586021094893341335458278360, 21.54558367742636199524882402519, 22.3292698933323292735498430646

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