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Properties

Label 1-927-927.425-r1-0-0
Degree $1$
Conductor $927$
Sign $-0.999 + 0.00831i$
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  + (0.153 + 0.988i)2-s + (−0.952 + 0.303i)4-s + (−0.969 − 0.243i)5-s + (−0.998 + 0.0615i)7-s + (−0.445 − 0.895i)8-s + (0.0922 − 0.995i)10-s + (0.153 + 0.988i)11-s + (0.552 + 0.833i)13-s + (−0.213 − 0.976i)14-s + (0.816 − 0.577i)16-s + (0.850 − 0.526i)17-s + (−0.982 + 0.183i)19-s + (0.998 − 0.0615i)20-s + (−0.952 + 0.303i)22-s + (0.153 − 0.988i)23-s + ⋯
L(s)  = 1  + (0.153 + 0.988i)2-s + (−0.952 + 0.303i)4-s + (−0.969 − 0.243i)5-s + (−0.998 + 0.0615i)7-s + (−0.445 − 0.895i)8-s + (0.0922 − 0.995i)10-s + (0.153 + 0.988i)11-s + (0.552 + 0.833i)13-s + (−0.213 − 0.976i)14-s + (0.816 − 0.577i)16-s + (0.850 − 0.526i)17-s + (−0.982 + 0.183i)19-s + (0.998 − 0.0615i)20-s + (−0.952 + 0.303i)22-s + (0.153 − 0.988i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.00831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.00831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $-0.999 + 0.00831i$
Analytic conductor: \(99.6199\)
Root analytic conductor: \(99.6199\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (425, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (1:\ ),\ -0.999 + 0.00831i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.003400811489 + 0.8178927536i\)
\(L(\frac12)\) \(\approx\) \(0.003400811489 + 0.8178927536i\)
\(L(1)\) \(\approx\) \(0.6355077774 + 0.4210661807i\)
\(L(1)\) \(\approx\) \(0.6355077774 + 0.4210661807i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 \)
good2 \( 1 + (0.153 + 0.988i)T \)
5 \( 1 + (-0.969 - 0.243i)T \)
7 \( 1 + (-0.998 + 0.0615i)T \)
11 \( 1 + (0.153 + 0.988i)T \)
13 \( 1 + (0.552 + 0.833i)T \)
17 \( 1 + (0.850 - 0.526i)T \)
19 \( 1 + (-0.982 + 0.183i)T \)
23 \( 1 + (0.153 - 0.988i)T \)
29 \( 1 + (0.696 - 0.717i)T \)
31 \( 1 + (-0.908 - 0.417i)T \)
37 \( 1 + (-0.602 - 0.798i)T \)
41 \( 1 + (0.696 + 0.717i)T \)
43 \( 1 + (0.992 + 0.122i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (0.982 - 0.183i)T \)
59 \( 1 + (0.998 + 0.0615i)T \)
61 \( 1 + (0.881 + 0.473i)T \)
67 \( 1 + (-0.998 - 0.0615i)T \)
71 \( 1 + (0.273 - 0.961i)T \)
73 \( 1 + (-0.273 + 0.961i)T \)
79 \( 1 + (-0.696 + 0.717i)T \)
83 \( 1 + (0.998 - 0.0615i)T \)
89 \( 1 + (-0.739 + 0.673i)T \)
97 \( 1 + (-0.0307 + 0.999i)T \)
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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

−21.3130136187676330652204256719, −20.35937248529230187102613149103, −19.50570696654711038175040416768, −19.21566308351857544517115066746, −18.50477671280727965814822252403, −17.45325231071535859147061054960, −16.46112574558177797952391343873, −15.65833799462975758974619239565, −14.81569367875258389121053713208, −13.88732894236343848019151231968, −13.00180781882633053475817082656, −12.42903213933229416299486011870, −11.567067301797128082840622725413, −10.66859337072824550282677800736, −10.27219966814653717201892778085, −8.93053141918289086407278293016, −8.44398247339682383079447853229, −7.3079034805646207113610452040, −6.11727806974288568802037867921, −5.322324239828954829545754495419, −3.86247265611715854891268668615, −3.529907518390999765889092246644, −2.71773077426036873394699430157, −1.13139150035212683302184094256, −0.259270135082351586027283932223, 0.82970345257268913625127670786, 2.64495301315194489257073436402, 3.9973887535872369631557111334, 4.22519606995060922451227442717, 5.49288142515354182892016365711, 6.51817255856760928841842484501, 7.11044554335355351958070137514, 7.97369185952078107376705978609, 8.91664279597977420621464827327, 9.53500082739268427669794640815, 10.606985612631868337919802186, 11.95358202351139715209639011672, 12.51408173487355820840185043778, 13.20534081979865520454757093549, 14.36223180339531233479560839680, 14.93042234873199757249074783045, 15.90994389727140437970621038432, 16.29807057749388298243297841190, 17.0027062844434666857328628251, 18.07602791279642002989204155716, 18.98546546732078493975744479915, 19.36319807011282610556595724463, 20.56224625141541389829607558649, 21.326173210591528573416918408, 22.5765691634076582636968500544

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