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Properties

Label 1-927-927.166-r0-0-0
Degree $1$
Conductor $927$
Sign $-0.256 - 0.966i$
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 + (−0.952 − 0.303i)7-s + (0.739 + 0.673i)8-s + (0.445 + 0.895i)10-s + (−0.696 + 0.717i)11-s + (−0.952 − 0.303i)13-s + (−0.0307 + 0.999i)14-s + (0.445 − 0.895i)16-s + (−0.153 + 0.988i)17-s + (−0.389 + 0.920i)19-s + (0.739 − 0.673i)20-s + (0.881 + 0.473i)22-s + (0.969 − 0.243i)23-s + ⋯
L(s)  = 1  + (−0.273 − 0.961i)2-s + (−0.850 + 0.526i)4-s + (−0.982 + 0.183i)5-s + (−0.952 − 0.303i)7-s + (0.739 + 0.673i)8-s + (0.445 + 0.895i)10-s + (−0.696 + 0.717i)11-s + (−0.952 − 0.303i)13-s + (−0.0307 + 0.999i)14-s + (0.445 − 0.895i)16-s + (−0.153 + 0.988i)17-s + (−0.389 + 0.920i)19-s + (0.739 − 0.673i)20-s + (0.881 + 0.473i)22-s + (0.969 − 0.243i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.256 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.256 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $-0.256 - 0.966i$
Analytic conductor: \(4.30496\)
Root analytic conductor: \(4.30496\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (166, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (0:\ ),\ -0.256 - 0.966i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2310389581 - 0.3003153111i\)
\(L(\frac12)\) \(\approx\) \(0.2310389581 - 0.3003153111i\)
\(L(1)\) \(\approx\) \(0.4869718309 - 0.1829170564i\)
\(L(1)\) \(\approx\) \(0.4869718309 - 0.1829170564i\)

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 \( 1 + (-0.952 - 0.303i)T \)
11 \( 1 + (-0.696 + 0.717i)T \)
13 \( 1 + (-0.952 - 0.303i)T \)
17 \( 1 + (-0.153 + 0.988i)T \)
19 \( 1 + (-0.389 + 0.920i)T \)
23 \( 1 + (0.969 - 0.243i)T \)
29 \( 1 + (-0.982 + 0.183i)T \)
31 \( 1 + (-0.998 + 0.0615i)T \)
37 \( 1 + (0.0922 - 0.995i)T \)
41 \( 1 + (0.650 - 0.759i)T \)
43 \( 1 + (0.0922 + 0.995i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (0.992 - 0.122i)T \)
59 \( 1 + (-0.952 + 0.303i)T \)
61 \( 1 + (-0.153 + 0.988i)T \)
67 \( 1 + (0.739 + 0.673i)T \)
71 \( 1 + (0.650 - 0.759i)T \)
73 \( 1 + (-0.982 - 0.183i)T \)
79 \( 1 + (0.332 - 0.943i)T \)
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

−22.32710511028168871844050024914, −21.57010070593863600402932399028, −20.230926712007032250998214599596, −19.47433044639961239939109468379, −18.8678969661713797310446067615, −18.30018858035085120837003592884, −16.964753149783428572831185807399, −16.56913835002209358647797574222, −15.6154685303392165920216858060, −15.33979306187969895051038234026, −14.30017694993032656373837210385, −13.22433626656344695497947775389, −12.75401614570245101252607975535, −11.55035313429718903171439005500, −10.718458719649361488001851420623, −9.45323794747497516669698130327, −9.08042546948123412837587530548, −8.0075923997510297968897924082, −7.24623377361291848513508181186, −6.606798008880591542035253922964, −5.391274585353231993537446326643, −4.7666560671307117219413682822, −3.61241530278496470226392396300, −2.623101215957584708367759192472, −0.65304669810694487626742931217, 0.3108921674464619394762430893, 1.91408840824895441098850906024, 2.930613654172768015968814848151, 3.75066227555419279251202946332, 4.47936827921828613859678153133, 5.63544899267917048276983248401, 7.15849303368099352761265363967, 7.61448562569662610814439669329, 8.65948601146860501593550968996, 9.579828572703345680305222028303, 10.45694182400073892876169589831, 10.8831185826975743746770484139, 12.1098433474750589816670811351, 12.7002455951421971464335783239, 13.10245120055293330597674576559, 14.58330374227426922792502299891, 15.09016984236502925579226911752, 16.31500224164878498073256687629, 16.880295898978987641450721623189, 17.8900024714453557339715424781, 18.75007555536756528465001856379, 19.40153457160763801884447232680, 19.90114042904185076979971112362, 20.650636690021905858396572296302, 21.571724176151305749605019600370

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