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Properties

Label 1-927-927.284-r0-0-0
Degree $1$
Conductor $927$
Sign $0.999 + 0.0301i$
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  + (−0.332 − 0.943i)2-s + (−0.779 + 0.626i)4-s + (−0.389 − 0.920i)5-s + (−0.850 + 0.526i)7-s + (0.850 + 0.526i)8-s + (−0.739 + 0.673i)10-s + (−0.982 − 0.183i)11-s + (0.881 + 0.473i)13-s + (0.779 + 0.626i)14-s + (0.213 − 0.976i)16-s + (0.696 − 0.717i)17-s + (−0.908 − 0.417i)19-s + (0.881 + 0.473i)20-s + (0.153 + 0.988i)22-s + (−0.650 − 0.759i)23-s + ⋯
L(s)  = 1  + (−0.332 − 0.943i)2-s + (−0.779 + 0.626i)4-s + (−0.389 − 0.920i)5-s + (−0.850 + 0.526i)7-s + (0.850 + 0.526i)8-s + (−0.739 + 0.673i)10-s + (−0.982 − 0.183i)11-s + (0.881 + 0.473i)13-s + (0.779 + 0.626i)14-s + (0.213 − 0.976i)16-s + (0.696 − 0.717i)17-s + (−0.908 − 0.417i)19-s + (0.881 + 0.473i)20-s + (0.153 + 0.988i)22-s + (−0.650 − 0.759i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0301i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0301i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $0.999 + 0.0301i$
Analytic conductor: \(4.30496\)
Root analytic conductor: \(4.30496\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (284, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (0:\ ),\ 0.999 + 0.0301i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5403375560 + 0.008158928417i\)
\(L(\frac12)\) \(\approx\) \(0.5403375560 + 0.008158928417i\)
\(L(1)\) \(\approx\) \(0.5717538106 - 0.2505440158i\)
\(L(1)\) \(\approx\) \(0.5717538106 - 0.2505440158i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 \)
good2 \( 1 + (-0.332 - 0.943i)T \)
5 \( 1 + (-0.389 - 0.920i)T \)
7 \( 1 + (-0.850 + 0.526i)T \)
11 \( 1 + (-0.982 - 0.183i)T \)
13 \( 1 + (0.881 + 0.473i)T \)
17 \( 1 + (0.696 - 0.717i)T \)
19 \( 1 + (-0.908 - 0.417i)T \)
23 \( 1 + (-0.650 - 0.759i)T \)
29 \( 1 + (-0.992 - 0.122i)T \)
31 \( 1 + (-0.213 + 0.976i)T \)
37 \( 1 + (-0.445 + 0.895i)T \)
41 \( 1 + (0.602 + 0.798i)T \)
43 \( 1 + (-0.552 + 0.833i)T \)
47 \( 1 + T \)
53 \( 1 + (0.816 - 0.577i)T \)
59 \( 1 + (0.850 + 0.526i)T \)
61 \( 1 + (0.969 + 0.243i)T \)
67 \( 1 + (-0.881 + 0.473i)T \)
71 \( 1 + (-0.389 + 0.920i)T \)
73 \( 1 + (0.602 + 0.798i)T \)
79 \( 1 + (-0.389 - 0.920i)T \)
83 \( 1 + (-0.881 - 0.473i)T \)
89 \( 1 + (0.932 - 0.361i)T \)
97 \( 1 + (0.969 - 0.243i)T \)
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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

−22.25069949440829863531549605825, −21.11647641161567354044921018037, −20.06150334881813835078060399958, −19.1285969612151103476843115154, −18.72562153005146696327070274283, −17.91694943235045105081152586353, −17.04179079345292626017389247426, −16.20932762926870589306237201690, −15.49843922726143488914226246067, −14.965147502149400093888263395302, −13.95920636295499980483169673058, −13.25540852849114576049735138545, −12.45508747303589465585461373953, −10.88750408676076235249672707557, −10.45167449155712355084226687195, −9.72259052795167091952219355118, −8.53256497967887733968886447646, −7.66846112222227270912083506784, −7.18488634055424550484813472652, −6.046942995080613535143317348716, −5.65235278916145525234785449328, −3.981317508383397357397682853952, −3.576447805068503903204085552560, −2.05115070480369800383204831561, −0.3506576620000937479892052797, 0.883306902928095061795548594821, 2.15045337688318540800796242030, 3.10756792149476175917346350035, 4.016630297290064380785658983383, 4.962334163592154432616414242822, 5.88307761400366951241074039936, 7.232599008882076759996056745548, 8.37824402634306636565784274092, 8.74946842962731796134525180094, 9.68678498098712648178680490005, 10.46895831438835623089958157494, 11.507618221607292812016119859831, 12.12344906204675775117635496936, 13.139417473402145021720038725124, 13.21525879361094693554785297519, 14.591677000473772644118523143004, 15.97554592189061091906319465989, 16.199447936669991176605570324452, 17.12987189622882868015322102585, 18.31097403547631083073894706324, 18.72775049115693857179323866966, 19.52586295048122449482913759684, 20.31753472335724195546132391972, 21.01685228133333992529230536503, 21.5459827888147923252957657715

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