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Properties

Label 1-927-927.248-r0-0-0
Degree $1$
Conductor $927$
Sign $0.774 + 0.632i$
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  + (−0.650 + 0.759i)2-s + (−0.153 − 0.988i)4-s + (0.992 + 0.122i)5-s + (−0.0307 − 0.999i)7-s + (0.850 + 0.526i)8-s + (−0.739 + 0.673i)10-s + (0.650 − 0.759i)11-s + (0.881 + 0.473i)13-s + (0.779 + 0.626i)14-s + (−0.952 + 0.303i)16-s + (0.273 + 0.961i)17-s + (0.0922 + 0.995i)19-s + (−0.0307 − 0.999i)20-s + (0.153 + 0.988i)22-s + (−0.650 − 0.759i)23-s + ⋯
L(s)  = 1  + (−0.650 + 0.759i)2-s + (−0.153 − 0.988i)4-s + (0.992 + 0.122i)5-s + (−0.0307 − 0.999i)7-s + (0.850 + 0.526i)8-s + (−0.739 + 0.673i)10-s + (0.650 − 0.759i)11-s + (0.881 + 0.473i)13-s + (0.779 + 0.626i)14-s + (−0.952 + 0.303i)16-s + (0.273 + 0.961i)17-s + (0.0922 + 0.995i)19-s + (−0.0307 − 0.999i)20-s + (0.153 + 0.988i)22-s + (−0.650 − 0.759i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.774 + 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.774 + 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $0.774 + 0.632i$
Analytic conductor: \(4.30496\)
Root analytic conductor: \(4.30496\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (248, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (0:\ ),\ 0.774 + 0.632i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.322685723 + 0.4716241753i\)
\(L(\frac12)\) \(\approx\) \(1.322685723 + 0.4716241753i\)
\(L(1)\) \(\approx\) \(0.9841630719 + 0.2508257385i\)
\(L(1)\) \(\approx\) \(0.9841630719 + 0.2508257385i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 \)
good2 \( 1 + (-0.650 + 0.759i)T \)
5 \( 1 + (0.992 + 0.122i)T \)
7 \( 1 + (-0.0307 - 0.999i)T \)
11 \( 1 + (0.650 - 0.759i)T \)
13 \( 1 + (0.881 + 0.473i)T \)
17 \( 1 + (0.273 + 0.961i)T \)
19 \( 1 + (0.0922 + 0.995i)T \)
23 \( 1 + (-0.650 - 0.759i)T \)
29 \( 1 + (0.389 + 0.920i)T \)
31 \( 1 + (-0.213 + 0.976i)T \)
37 \( 1 + (-0.445 + 0.895i)T \)
41 \( 1 + (0.389 - 0.920i)T \)
43 \( 1 + (0.998 + 0.0615i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (0.0922 + 0.995i)T \)
59 \( 1 + (0.0307 - 0.999i)T \)
61 \( 1 + (0.969 + 0.243i)T \)
67 \( 1 + (0.0307 - 0.999i)T \)
71 \( 1 + (-0.602 - 0.798i)T \)
73 \( 1 + (0.602 + 0.798i)T \)
79 \( 1 + (-0.389 - 0.920i)T \)
83 \( 1 + (0.0307 + 0.999i)T \)
89 \( 1 + (0.932 - 0.361i)T \)
97 \( 1 + (-0.696 - 0.717i)T \)
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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

−21.58490871398057500457615129160, −20.965215601542560526010356897534, −20.25811491914870700218099716103, −19.41797272038823421905986296085, −18.50338786929500162880579687250, −17.78744587315088105062389636682, −17.525121277717627182594163020694, −16.320172679389716766312754879916, −15.62547964717436253463000536931, −14.49635941900532376929033040117, −13.450974277665182622147794954011, −12.937984995204612065509344795316, −11.89266636482363023905731350786, −11.39227456024938083516306174899, −10.22016073321636723533712932123, −9.46348487285989995194055571654, −9.07190517248022331568418520512, −8.0684916851600124917677703585, −6.9880724374636003161309501752, −5.97484578560009693596273716966, −5.04128881058604261450886907846, −3.87170334236700563356917275951, −2.669521563284981860263639697664, −2.076521752970348986915896379604, −0.9690335554259694879747605206, 1.13065419212995142782305539038, 1.70819977037064243424310516409, 3.41595216206382054488445973526, 4.416720006457507867201064276876, 5.67870032215298374756354652196, 6.30303129878947075574896147867, 6.91754860796862690338612194289, 8.12606252765739233164884129584, 8.77483574162166054267005951405, 9.687902815950178862703039354740, 10.57174791238022152439301168926, 10.88869298203491358666510703597, 12.40972452216533889424179589142, 13.53056069327043854948259121662, 14.23751187780685485631836695217, 14.43337910697085126609615790661, 15.94849649455885891947228621135, 16.524396772208835047815687864171, 17.13106979497740703286051697606, 17.8477377362442184542466172014, 18.69169466049700194975418281527, 19.34640962044009618016501048789, 20.32852608553645667627738737141, 21.05069115142182007730563923130, 22.05910873383775949887045506541

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