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Properties

Label 1-927-927.214-r0-0-0
Degree $1$
Conductor $927$
Sign $0.349 + 0.936i$
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  + (−0.779 − 0.626i)2-s + (0.213 + 0.976i)4-s + (−0.696 − 0.717i)5-s + (0.552 − 0.833i)7-s + (0.445 − 0.895i)8-s + (0.0922 + 0.995i)10-s + (−0.779 − 0.626i)11-s + (−0.998 − 0.0615i)13-s + (−0.952 + 0.303i)14-s + (−0.908 + 0.417i)16-s + (−0.850 − 0.526i)17-s + (−0.982 − 0.183i)19-s + (0.552 − 0.833i)20-s + (0.213 + 0.976i)22-s + (−0.779 + 0.626i)23-s + ⋯
L(s)  = 1  + (−0.779 − 0.626i)2-s + (0.213 + 0.976i)4-s + (−0.696 − 0.717i)5-s + (0.552 − 0.833i)7-s + (0.445 − 0.895i)8-s + (0.0922 + 0.995i)10-s + (−0.779 − 0.626i)11-s + (−0.998 − 0.0615i)13-s + (−0.952 + 0.303i)14-s + (−0.908 + 0.417i)16-s + (−0.850 − 0.526i)17-s + (−0.982 − 0.183i)19-s + (0.552 − 0.833i)20-s + (0.213 + 0.976i)22-s + (−0.779 + 0.626i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.349 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.349 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $0.349 + 0.936i$
Analytic conductor: \(4.30496\)
Root analytic conductor: \(4.30496\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (214, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (0:\ ),\ 0.349 + 0.936i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07699897112 + 0.05343950926i\)
\(L(\frac12)\) \(\approx\) \(0.07699897112 + 0.05343950926i\)
\(L(1)\) \(\approx\) \(0.4469363682 - 0.2267278883i\)
\(L(1)\) \(\approx\) \(0.4469363682 - 0.2267278883i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 \)
good2 \( 1 + (-0.779 - 0.626i)T \)
5 \( 1 + (-0.696 - 0.717i)T \)
7 \( 1 + (0.552 - 0.833i)T \)
11 \( 1 + (-0.779 - 0.626i)T \)
13 \( 1 + (-0.998 - 0.0615i)T \)
17 \( 1 + (-0.850 - 0.526i)T \)
19 \( 1 + (-0.982 - 0.183i)T \)
23 \( 1 + (-0.779 + 0.626i)T \)
29 \( 1 + (0.969 - 0.243i)T \)
31 \( 1 + (0.816 + 0.577i)T \)
37 \( 1 + (-0.602 + 0.798i)T \)
41 \( 1 + (0.969 + 0.243i)T \)
43 \( 1 + (-0.389 + 0.920i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (-0.982 - 0.183i)T \)
59 \( 1 + (0.552 + 0.833i)T \)
61 \( 1 + (-0.0307 + 0.999i)T \)
67 \( 1 + (0.552 + 0.833i)T \)
71 \( 1 + (-0.273 - 0.961i)T \)
73 \( 1 + (-0.273 - 0.961i)T \)
79 \( 1 + (0.969 - 0.243i)T \)
83 \( 1 + (0.552 - 0.833i)T \)
89 \( 1 + (0.739 + 0.673i)T \)
97 \( 1 + (0.881 + 0.473i)T \)
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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

−21.81759570699310080233915825542, −20.78407462126756611438974602891, −19.82873964541911734395503061434, −19.16260685814836818673022182618, −18.52695440188247594593427605485, −17.71943957597105350680536176574, −17.23348483697167326911788159259, −15.77888026458443456412863589257, −15.603340099562579660029956540984, −14.666057649647393529419696923852, −14.266127028913224855215668400274, −12.69465663113040095631244093013, −11.939583442034927262040787990669, −10.917433254277836669446838321983, −10.38241444086808990146600544200, −9.406921393385605971394649837335, −8.26847902991864883802466640765, −7.96074271131754968364972971043, −6.8912254770587827744315525523, −6.210022924197911089061938783810, −5.03512039560828470726418286217, −4.295483408843048310386496640693, −2.503118398845929202965763160892, −2.08275794812960810547591768189, −0.06120178505007766587058159705, 0.99007216399497149035483607198, 2.20461366223894154955632871291, 3.25786124777076343198055343863, 4.39184102470911129406539568225, 4.8898759374859614072373010169, 6.58809279274289466060575996131, 7.58693438462764821698347333283, 8.12999123064029240222577752065, 8.838261755226543410920642729659, 9.94836359143118500076212874351, 10.65031900271187467451100119440, 11.50318831727073661317575038430, 12.096620467692412000902056194948, 13.12032164973257987111781530140, 13.69109780236337806624171011989, 15.03800236992785125478618114103, 15.9887480707290768157045242993, 16.51806727386604992152446182112, 17.50375442166433657286687324699, 17.86137369192948382082809065761, 19.21470636681343199239304042835, 19.568904703280621164763114650220, 20.31644094442582850200349446521, 21.061000918308407966608615290948, 21.650764410099274648659043662667

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