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Properties

Label 1-927-927.265-r0-0-0
Degree $1$
Conductor $927$
Sign $0.483 - 0.875i$
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  + (−0.779 − 0.626i)2-s + (0.213 + 0.976i)4-s + (−0.696 − 0.717i)5-s + (0.445 + 0.895i)7-s + (0.445 − 0.895i)8-s + (0.0922 + 0.995i)10-s + (0.932 − 0.361i)11-s + (0.552 − 0.833i)13-s + (0.213 − 0.976i)14-s + (−0.908 + 0.417i)16-s + (−0.0307 + 0.999i)17-s + (0.650 − 0.759i)19-s + (0.552 − 0.833i)20-s + (−0.952 − 0.303i)22-s + (−0.153 − 0.988i)23-s + ⋯
L(s)  = 1  + (−0.779 − 0.626i)2-s + (0.213 + 0.976i)4-s + (−0.696 − 0.717i)5-s + (0.445 + 0.895i)7-s + (0.445 − 0.895i)8-s + (0.0922 + 0.995i)10-s + (0.932 − 0.361i)11-s + (0.552 − 0.833i)13-s + (0.213 − 0.976i)14-s + (−0.908 + 0.417i)16-s + (−0.0307 + 0.999i)17-s + (0.650 − 0.759i)19-s + (0.552 − 0.833i)20-s + (−0.952 − 0.303i)22-s + (−0.153 − 0.988i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.483 - 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.483 - 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $0.483 - 0.875i$
Analytic conductor: \(4.30496\)
Root analytic conductor: \(4.30496\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (265, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (0:\ ),\ 0.483 - 0.875i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8543258115 - 0.5039501186i\)
\(L(\frac12)\) \(\approx\) \(0.8543258115 - 0.5039501186i\)
\(L(1)\) \(\approx\) \(0.7386568335 - 0.2502179771i\)
\(L(1)\) \(\approx\) \(0.7386568335 - 0.2502179771i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 \)
good2 \( 1 + (-0.779 - 0.626i)T \)
5 \( 1 + (-0.696 - 0.717i)T \)
7 \( 1 + (0.445 + 0.895i)T \)
11 \( 1 + (0.932 - 0.361i)T \)
13 \( 1 + (0.552 - 0.833i)T \)
17 \( 1 + (-0.0307 + 0.999i)T \)
19 \( 1 + (0.650 - 0.759i)T \)
23 \( 1 + (-0.153 - 0.988i)T \)
29 \( 1 + (0.969 - 0.243i)T \)
31 \( 1 + (-0.908 + 0.417i)T \)
37 \( 1 + (-0.602 + 0.798i)T \)
41 \( 1 + (-0.273 - 0.961i)T \)
43 \( 1 + (-0.389 + 0.920i)T \)
47 \( 1 + T \)
53 \( 1 + (0.332 + 0.943i)T \)
59 \( 1 + (0.445 - 0.895i)T \)
61 \( 1 + (0.881 - 0.473i)T \)
67 \( 1 + (0.552 + 0.833i)T \)
71 \( 1 + (-0.696 + 0.717i)T \)
73 \( 1 + (-0.273 - 0.961i)T \)
79 \( 1 + (-0.696 - 0.717i)T \)
83 \( 1 + (0.552 - 0.833i)T \)
89 \( 1 + (0.739 + 0.673i)T \)
97 \( 1 + (0.881 + 0.473i)T \)
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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

−22.26756433859978304767752626094, −20.98587344494204230416473615695, −20.06599816018422914389513029897, −19.6382116117569969907508009795, −18.66361186613376526250236859170, −18.08765753980948732740938170091, −17.27196387961828953446972214921, −16.38300666030845385654318749568, −15.87255085867360112515844137944, −14.80331138690809633007881136338, −14.213631613098023670108529678182, −13.66248350523730842920527094023, −11.836178202334080753210710364470, −11.478486982168528904557761829317, −10.54936876821967881122695163275, −9.75208182144853082370289310281, −8.86597647666597144773829406973, −7.86833652372141310004433065710, −7.127149701563859829132217949430, −6.73105985136896981493441078642, −5.49062247055560009438558234743, −4.322116273944089099293450897927, −3.5431442639054050575584414707, −1.9665624320862676574668667135, −0.95844893188789707580089335331, 0.78592239696121190551881562366, 1.69675282910393234496152612067, 2.95861765021770755123165670325, 3.82202590666214275020744743052, 4.803000328951212134768367018, 5.96408823075058577662519827883, 7.094445660667870993246732162533, 8.3083040645284420723199590412, 8.51270857584240229347018605908, 9.28816964848584475581849734100, 10.47544781229354447849463981663, 11.26319846759755637189566790390, 12.01076367851825151166093510563, 12.523030801999981373636832002552, 13.43774370522628130483708118238, 14.71194782496701873776266864342, 15.65148270106419517699578295019, 16.18080778074036447855610970337, 17.222613619324043436645459859988, 17.74287773720757461806749020653, 18.76923113110513406454250646052, 19.304842140516899496499711933616, 20.23177104544947545962241042452, 20.58816402546518825629467729786, 21.771538382543578306811752859490

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