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Properties

Label 1-828-828.263-r1-0-0
Degree $1$
Conductor $828$
Sign $-0.790 - 0.612i$
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  + (0.981 − 0.189i)5-s + (0.0475 − 0.998i)7-s + (0.786 − 0.618i)11-s + (0.0475 + 0.998i)13-s + (−0.959 + 0.281i)17-s + (−0.959 − 0.281i)19-s + (0.928 − 0.371i)25-s + (−0.723 − 0.690i)29-s + (−0.580 − 0.814i)31-s + (−0.142 − 0.989i)35-s + (0.654 − 0.755i)37-s + (−0.981 + 0.189i)41-s + (0.580 − 0.814i)43-s + (−0.5 − 0.866i)47-s + (−0.995 − 0.0950i)49-s + ⋯
L(s)  = 1  + (0.981 − 0.189i)5-s + (0.0475 − 0.998i)7-s + (0.786 − 0.618i)11-s + (0.0475 + 0.998i)13-s + (−0.959 + 0.281i)17-s + (−0.959 − 0.281i)19-s + (0.928 − 0.371i)25-s + (−0.723 − 0.690i)29-s + (−0.580 − 0.814i)31-s + (−0.142 − 0.989i)35-s + (0.654 − 0.755i)37-s + (−0.981 + 0.189i)41-s + (0.580 − 0.814i)43-s + (−0.5 − 0.866i)47-s + (−0.995 − 0.0950i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(828\)    =    \(2^{2} \cdot 3^{2} \cdot 23\)
Sign: $-0.790 - 0.612i$
Analytic conductor: \(88.9809\)
Root analytic conductor: \(88.9809\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{828} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 828,\ (1:\ ),\ -0.790 - 0.612i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4934553548 - 1.441428225i\)
\(L(\frac12)\) \(\approx\) \(0.4934553548 - 1.441428225i\)
\(L(1)\) \(\approx\) \(1.095940593 - 0.3164087812i\)
\(L(1)\) \(\approx\) \(1.095940593 - 0.3164087812i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
23 \( 1 \)
good5 \( 1 + (0.981 - 0.189i)T \)
7 \( 1 + (0.0475 - 0.998i)T \)
11 \( 1 + (0.786 - 0.618i)T \)
13 \( 1 + (0.0475 + 0.998i)T \)
17 \( 1 + (-0.959 + 0.281i)T \)
19 \( 1 + (-0.959 - 0.281i)T \)
29 \( 1 + (-0.723 - 0.690i)T \)
31 \( 1 + (-0.580 - 0.814i)T \)
37 \( 1 + (0.654 - 0.755i)T \)
41 \( 1 + (-0.981 + 0.189i)T \)
43 \( 1 + (0.580 - 0.814i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (0.841 + 0.540i)T \)
59 \( 1 + (0.0475 + 0.998i)T \)
61 \( 1 + (0.995 - 0.0950i)T \)
67 \( 1 + (-0.786 - 0.618i)T \)
71 \( 1 + (-0.142 + 0.989i)T \)
73 \( 1 + (-0.959 - 0.281i)T \)
79 \( 1 + (-0.888 - 0.458i)T \)
83 \( 1 + (-0.981 - 0.189i)T \)
89 \( 1 + (0.415 + 0.909i)T \)
97 \( 1 + (0.327 - 0.945i)T \)
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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

−22.22121898320062649429759925717, −21.65340948618207149037511964335, −20.71134018197487676749965285960, −19.99486233439705276927829276453, −19.00568721068590608781123175390, −18.07982327170215836739406765039, −17.685656213166693743561491498186, −16.81337906666288504600095200214, −15.75397495542835926981791645013, −14.84134378600172324135991367324, −14.45572660856005561321981109098, −13.112578525796086704368578060801, −12.74920930609049245984917818601, −11.66145462244509868428949477263, −10.748237219587919790282074021046, −9.853773986087889906547542555311, −9.078940663120570338720298759205, −8.38792820182108792972853088897, −7.03909644139553389701328008162, −6.27220227414920643667490684680, −5.47550509563468698554637042927, −4.57626414123999588367522929648, −3.20353804107795678819887840999, −2.26056335404899773837117762387, −1.454196314391801928574638717282, 0.30368375619490168504427414461, 1.520844765150775334109162260964, 2.33376212775687554498695769170, 3.88127455063098931836121742181, 4.41106418423585434725237485700, 5.74628721575873727179854500630, 6.50800485771860720984000096583, 7.22391876125196536014371535548, 8.610127701933178408948098599131, 9.166236563890459651006376138318, 10.11884511495252337964745623888, 10.964337722743878756486402620424, 11.67957354299585750174573673967, 13.05466314090101820003957348134, 13.45926946923623735326155533828, 14.25939288714530659846901500704, 15.00808427219303698715688049630, 16.361058383179401357779520732330, 16.92191816349398698920425207411, 17.40333139088387673769440465886, 18.44638148887597012649637209639, 19.33659602949773957567466389676, 20.08769144068359149807883313858, 20.92014695306013447356932144791, 21.68727843170227317063464618989

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