Properties

Label 1-927-927.14-r1-0-0
Degree $1$
Conductor $927$
Sign $0.980 - 0.196i$
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  + (−0.969 − 0.243i)2-s + (0.881 + 0.473i)4-s + (−0.650 − 0.759i)5-s + (0.213 + 0.976i)7-s + (−0.739 − 0.673i)8-s + (0.445 + 0.895i)10-s + (−0.969 − 0.243i)11-s + (−0.952 − 0.303i)13-s + (0.0307 − 0.999i)14-s + (0.552 + 0.833i)16-s + (−0.932 + 0.361i)17-s + (−0.602 − 0.798i)19-s + (−0.213 − 0.976i)20-s + (0.881 + 0.473i)22-s + (−0.969 + 0.243i)23-s + ⋯
L(s)  = 1  + (−0.969 − 0.243i)2-s + (0.881 + 0.473i)4-s + (−0.650 − 0.759i)5-s + (0.213 + 0.976i)7-s + (−0.739 − 0.673i)8-s + (0.445 + 0.895i)10-s + (−0.969 − 0.243i)11-s + (−0.952 − 0.303i)13-s + (0.0307 − 0.999i)14-s + (0.552 + 0.833i)16-s + (−0.932 + 0.361i)17-s + (−0.602 − 0.798i)19-s + (−0.213 − 0.976i)20-s + (0.881 + 0.473i)22-s + (−0.969 + 0.243i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.980 - 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.980 - 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $0.980 - 0.196i$
Analytic conductor: \(99.6199\)
Root analytic conductor: \(99.6199\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (1:\ ),\ 0.980 - 0.196i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2536572528 + 0.02511387255i\)
\(L(\frac12)\) \(\approx\) \(0.2536572528 + 0.02511387255i\)
\(L(1)\) \(\approx\) \(0.4240486270 - 0.05952813513i\)
\(L(1)\) \(\approx\) \(0.4240486270 - 0.05952813513i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 \)
good2 \( 1 + (-0.969 - 0.243i)T \)
5 \( 1 + (-0.650 - 0.759i)T \)
7 \( 1 + (0.213 + 0.976i)T \)
11 \( 1 + (-0.969 - 0.243i)T \)
13 \( 1 + (-0.952 - 0.303i)T \)
17 \( 1 + (-0.932 + 0.361i)T \)
19 \( 1 + (-0.602 - 0.798i)T \)
23 \( 1 + (-0.969 + 0.243i)T \)
29 \( 1 + (-0.332 + 0.943i)T \)
31 \( 1 + (-0.998 + 0.0615i)T \)
37 \( 1 + (0.0922 - 0.995i)T \)
41 \( 1 + (-0.332 - 0.943i)T \)
43 \( 1 + (-0.908 - 0.417i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (0.602 + 0.798i)T \)
59 \( 1 + (-0.213 + 0.976i)T \)
61 \( 1 + (-0.153 + 0.988i)T \)
67 \( 1 + (0.213 - 0.976i)T \)
71 \( 1 + (0.982 + 0.183i)T \)
73 \( 1 + (-0.982 - 0.183i)T \)
79 \( 1 + (0.332 - 0.943i)T \)
83 \( 1 + (-0.213 - 0.976i)T \)
89 \( 1 + (0.850 + 0.526i)T \)
97 \( 1 + (-0.779 + 0.626i)T \)
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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

−21.612546906075028114785242155817, −20.482806457810765789563409520495, −20.0712133404582074989249470012, −19.220663382072196366431844486526, −18.48338684015559415192020788354, −17.84212491900245213859798054949, −16.96369300549843602504434953830, −16.259011578606321174703139129899, −15.38911693484735714091694097055, −14.73784117755505452993043030784, −13.9570975201781249407599323825, −12.74671894831964118595011947477, −11.634793517874439066172146997187, −11.04556133901801857659488240934, −10.1881146743516314576841996321, −9.73368721106365169049852521635, −8.23160671308510002867825257979, −7.80710158350962134720223871176, −7.008504462696494849844614228957, −6.32590413133574362890767592870, −4.94714755804690322649826357366, −3.94781512286973028750986295382, −2.685733858771637144033256261791, −1.85931944344354151533873585772, −0.2402896494060495354161008199, 0.28143328497171986331177406132, 1.86160096249420773589611054729, 2.54081236026479864554349160584, 3.75385760266329320730886024972, 4.98379645341334386117922233756, 5.79043376805698905456581901569, 7.13568335164402817148682077649, 7.81816968831760240792622179006, 8.797139110102500896813425791152, 9.01112876299381366223393837929, 10.319866214197903906932577755774, 11.0401296898006992980066178013, 11.964591949469141981119749536991, 12.51992349068255857164967981433, 13.2797749269086363692282474333, 14.95592957883856278192617836925, 15.42098459177563061594738249175, 16.13509259427631007222976092862, 16.94509622357606676623948499801, 17.84815067523461791393116837559, 18.409700208927750830258896842791, 19.3873055247857997749564559120, 19.91075018609455003168626638686, 20.606679607131624585529411461734, 21.70029806701945610658848464535

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