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Properties

Label 1-927-927.232-r0-0-0
Degree $1$
Conductor $927$
Sign $0.503 + 0.863i$
Analytic cond. $4.30496$
Root an. cond. $4.30496$
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  + (−0.273 − 0.961i)2-s + (−0.850 + 0.526i)4-s + (−0.982 + 0.183i)5-s + (0.213 + 0.976i)7-s + (0.739 + 0.673i)8-s + (0.445 + 0.895i)10-s + (0.969 + 0.243i)11-s + (0.213 + 0.976i)13-s + (0.881 − 0.473i)14-s + (0.445 − 0.895i)16-s + (−0.779 − 0.626i)17-s + (0.992 − 0.122i)19-s + (0.739 − 0.673i)20-s + (−0.0307 − 0.999i)22-s + (−0.696 − 0.717i)23-s + ⋯
L(s)  = 1  + (−0.273 − 0.961i)2-s + (−0.850 + 0.526i)4-s + (−0.982 + 0.183i)5-s + (0.213 + 0.976i)7-s + (0.739 + 0.673i)8-s + (0.445 + 0.895i)10-s + (0.969 + 0.243i)11-s + (0.213 + 0.976i)13-s + (0.881 − 0.473i)14-s + (0.445 − 0.895i)16-s + (−0.779 − 0.626i)17-s + (0.992 − 0.122i)19-s + (0.739 − 0.673i)20-s + (−0.0307 − 0.999i)22-s + (−0.696 − 0.717i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.503 + 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.503 + 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $0.503 + 0.863i$
Analytic conductor: \(4.30496\)
Root analytic conductor: \(4.30496\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (232, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (0:\ ),\ 0.503 + 0.863i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6585338415 + 0.3782538516i\)
\(L(\frac12)\) \(\approx\) \(0.6585338415 + 0.3782538516i\)
\(L(1)\) \(\approx\) \(0.7337303716 - 0.07100039003i\)
\(L(1)\) \(\approx\) \(0.7337303716 - 0.07100039003i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 \)
good2 \( 1 + (-0.273 - 0.961i)T \)
5 \( 1 + (-0.982 + 0.183i)T \)
7 \( 1 + (0.213 + 0.976i)T \)
11 \( 1 + (0.969 + 0.243i)T \)
13 \( 1 + (0.213 + 0.976i)T \)
17 \( 1 + (-0.779 - 0.626i)T \)
19 \( 1 + (0.992 - 0.122i)T \)
23 \( 1 + (-0.696 - 0.717i)T \)
29 \( 1 + (-0.982 + 0.183i)T \)
31 \( 1 + (0.552 + 0.833i)T \)
37 \( 1 + (0.0922 - 0.995i)T \)
41 \( 1 + (0.332 + 0.943i)T \)
43 \( 1 + (0.0922 + 0.995i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (-0.389 + 0.920i)T \)
59 \( 1 + (0.213 - 0.976i)T \)
61 \( 1 + (-0.779 - 0.626i)T \)
67 \( 1 + (0.739 + 0.673i)T \)
71 \( 1 + (0.332 + 0.943i)T \)
73 \( 1 + (-0.982 - 0.183i)T \)
79 \( 1 + (0.650 + 0.759i)T \)
83 \( 1 + (0.739 - 0.673i)T \)
89 \( 1 + (-0.850 - 0.526i)T \)
97 \( 1 + (0.932 + 0.361i)T \)
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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

−22.31715648965587470392686994867, −20.68381649847448579111693751175, −19.91500567931339814934229786705, −19.46131769119309712214272611178, −18.45134187535151758760005859627, −17.54599041520011960175747034030, −16.95409185766645670618448738194, −16.20439891161029554245071787500, −15.37265734561267884919328970252, −14.83383387247798670990251906267, −13.754042321317870873685719011164, −13.253812448312309147844340838117, −12.02636930671493206319895131985, −11.13910849875074616390814710472, −10.27334604955510541886409617768, −9.31812156165062776513484632176, −8.34564521225055851504119195588, −7.747249313126959944034579980791, −7.0399359808781258914629410063, −6.07453562284028306616020750143, −5.05847135120719026663987686641, −3.9947664788948614530516877209, −3.60338362710296684193079833599, −1.456397109805445399397405977631, −0.432741875240373319836115242469, 1.25245850623583466302644513060, 2.3190447089929369189066143407, 3.25711913015626466375217925886, 4.241073306309023471201451051585, 4.86540552082263247757710067396, 6.316116405877489436959001158356, 7.34954118836406767610163966777, 8.29370454847484342067926934142, 9.10381618632112426089313465827, 9.59871258944391333850692848785, 11.06234065688866105163429626785, 11.46786527630997120031572015592, 12.0932760442475974119369958875, 12.81041708866912307897225026569, 14.12530524394379118660895317163, 14.556413239774493630543486492100, 15.81752332036348655234547589001, 16.36211583061557390581999452690, 17.55842295327346501953832818588, 18.31685468352934248601693013720, 18.85444283861515850058752690250, 19.72071809493626456965866703116, 20.201769339618766595260674188597, 21.16512585822073889206565252994, 22.05385247600373525920523191207

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