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Properties

Label 1-927-927.545-r1-0-0
Degree $1$
Conductor $927$
Sign $0.852 + 0.523i$
Analytic cond. $99.6199$
Root an. cond. $99.6199$
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  + (0.908 + 0.417i)2-s + (0.650 + 0.759i)4-s + (0.998 − 0.0615i)5-s + (−0.696 − 0.717i)7-s + (0.273 + 0.961i)8-s + (0.932 + 0.361i)10-s + (0.908 + 0.417i)11-s + (0.969 − 0.243i)13-s + (−0.332 − 0.943i)14-s + (−0.153 + 0.988i)16-s + (0.602 + 0.798i)17-s + (0.739 − 0.673i)19-s + (0.696 + 0.717i)20-s + (0.650 + 0.759i)22-s + (0.908 − 0.417i)23-s + ⋯
L(s)  = 1  + (0.908 + 0.417i)2-s + (0.650 + 0.759i)4-s + (0.998 − 0.0615i)5-s + (−0.696 − 0.717i)7-s + (0.273 + 0.961i)8-s + (0.932 + 0.361i)10-s + (0.908 + 0.417i)11-s + (0.969 − 0.243i)13-s + (−0.332 − 0.943i)14-s + (−0.153 + 0.988i)16-s + (0.602 + 0.798i)17-s + (0.739 − 0.673i)19-s + (0.696 + 0.717i)20-s + (0.650 + 0.759i)22-s + (0.908 − 0.417i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.852 + 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.852 + 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $0.852 + 0.523i$
Analytic conductor: \(99.6199\)
Root analytic conductor: \(99.6199\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (545, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (1:\ ),\ 0.852 + 0.523i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.347427377 + 1.511294658i\)
\(L(\frac12)\) \(\approx\) \(5.347427377 + 1.511294658i\)
\(L(1)\) \(\approx\) \(2.337252581 + 0.5321910537i\)
\(L(1)\) \(\approx\) \(2.337252581 + 0.5321910537i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 \)
good2 \( 1 + (0.908 + 0.417i)T \)
5 \( 1 + (0.998 - 0.0615i)T \)
7 \( 1 + (-0.696 - 0.717i)T \)
11 \( 1 + (0.908 + 0.417i)T \)
13 \( 1 + (0.969 - 0.243i)T \)
17 \( 1 + (0.602 + 0.798i)T \)
19 \( 1 + (0.739 - 0.673i)T \)
23 \( 1 + (0.908 - 0.417i)T \)
29 \( 1 + (-0.552 - 0.833i)T \)
31 \( 1 + (-0.779 - 0.626i)T \)
37 \( 1 + (-0.850 - 0.526i)T \)
41 \( 1 + (-0.552 + 0.833i)T \)
43 \( 1 + (-0.0307 - 0.999i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (-0.739 + 0.673i)T \)
59 \( 1 + (0.696 - 0.717i)T \)
61 \( 1 + (0.992 - 0.122i)T \)
67 \( 1 + (-0.696 + 0.717i)T \)
71 \( 1 + (-0.445 - 0.895i)T \)
73 \( 1 + (0.445 + 0.895i)T \)
79 \( 1 + (0.552 + 0.833i)T \)
83 \( 1 + (0.696 + 0.717i)T \)
89 \( 1 + (0.982 + 0.183i)T \)
97 \( 1 + (-0.389 - 0.920i)T \)
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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

−21.67387409248906024610738098157, −20.9141205193478444390320374545, −20.34356136541562465788168433657, −19.12414169159515785482442694851, −18.74253893307017833728025714409, −17.79222113580471988476668817897, −16.4283119086254391830192315544, −16.175948649268614774984359753739, −14.95678112529507808139461164363, −14.21652163519799344821915295296, −13.61142833412045766662483450780, −12.832809264081203203845484664904, −12.01960177065020491655459649793, −11.24910713109634930509458092029, −10.30524053015835831509183640624, −9.39311400060299487551990494762, −8.914750086292917227314122516761, −7.16520324891296530755184044777, −6.386260599804122479895993627511, −5.68501132205808838441490181338, −5.049875594480717056031797425347, −3.477308755644951900203931017202, −3.15904955861320796928141841523, −1.8208781457296661115841487480, −1.086024991227479402133871978598, 1.01120052794590813432275204771, 2.10383005316080989105372564531, 3.347579566837714998038282440616, 3.9457372532598583664351917598, 5.133768622825185845745700082631, 5.970149999550287420950745458330, 6.65986858377115671692898667718, 7.36922132834940372015053489405, 8.62428080907969821539397603547, 9.50438344661713143607063292780, 10.463527330253394629413640955301, 11.28972549842835431903085280115, 12.40484259688891179475221134135, 13.151068706791499026289874933898, 13.62790471868450283461364051197, 14.44414102131756184639443866104, 15.23315230093514128729949774780, 16.20434400057522624939321529003, 17.02397457266190759124370096317, 17.3171558699570486754996201198, 18.50088084790200572988562245766, 19.60319176520180115267662543092, 20.50618899678005758074805847400, 20.899255618343831051670914965720, 22.05552472487293791766290254399

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