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Properties

Label 1-927-927.760-r1-0-0
Degree $1$
Conductor $927$
Sign $0.467 - 0.883i$
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  + (0.213 + 0.976i)2-s + (−0.908 + 0.417i)4-s + (0.0307 − 0.999i)5-s + (−0.389 − 0.920i)7-s + (−0.602 − 0.798i)8-s + (0.982 − 0.183i)10-s + (−0.213 − 0.976i)11-s + (0.992 + 0.122i)13-s + (0.816 − 0.577i)14-s + (0.650 − 0.759i)16-s + (0.445 + 0.895i)17-s + (0.932 + 0.361i)19-s + (0.389 + 0.920i)20-s + (0.908 − 0.417i)22-s + (0.213 − 0.976i)23-s + ⋯
L(s)  = 1  + (0.213 + 0.976i)2-s + (−0.908 + 0.417i)4-s + (0.0307 − 0.999i)5-s + (−0.389 − 0.920i)7-s + (−0.602 − 0.798i)8-s + (0.982 − 0.183i)10-s + (−0.213 − 0.976i)11-s + (0.992 + 0.122i)13-s + (0.816 − 0.577i)14-s + (0.650 − 0.759i)16-s + (0.445 + 0.895i)17-s + (0.932 + 0.361i)19-s + (0.389 + 0.920i)20-s + (0.908 − 0.417i)22-s + (0.213 − 0.976i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.467 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.467 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $0.467 - 0.883i$
Analytic conductor: \(99.6199\)
Root analytic conductor: \(99.6199\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (760, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (1:\ ),\ 0.467 - 0.883i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.621677776 - 0.9763332677i\)
\(L(\frac12)\) \(\approx\) \(1.621677776 - 0.9763332677i\)
\(L(1)\) \(\approx\) \(1.107854687 + 0.06633326322i\)
\(L(1)\) \(\approx\) \(1.107854687 + 0.06633326322i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 \)
good2 \( 1 + (0.213 + 0.976i)T \)
5 \( 1 + (0.0307 - 0.999i)T \)
7 \( 1 + (-0.389 - 0.920i)T \)
11 \( 1 + (-0.213 - 0.976i)T \)
13 \( 1 + (0.992 + 0.122i)T \)
17 \( 1 + (0.445 + 0.895i)T \)
19 \( 1 + (0.932 + 0.361i)T \)
23 \( 1 + (0.213 - 0.976i)T \)
29 \( 1 + (0.881 - 0.473i)T \)
31 \( 1 + (-0.332 - 0.943i)T \)
37 \( 1 + (0.273 + 0.961i)T \)
41 \( 1 + (0.881 + 0.473i)T \)
43 \( 1 + (0.696 + 0.717i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (-0.932 - 0.361i)T \)
59 \( 1 + (-0.389 + 0.920i)T \)
61 \( 1 + (-0.998 - 0.0615i)T \)
67 \( 1 + (0.389 - 0.920i)T \)
71 \( 1 + (0.850 - 0.526i)T \)
73 \( 1 + (0.850 - 0.526i)T \)
79 \( 1 + (0.881 - 0.473i)T \)
83 \( 1 + (-0.389 - 0.920i)T \)
89 \( 1 + (-0.0922 - 0.995i)T \)
97 \( 1 + (0.552 + 0.833i)T \)
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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

−21.74774332696683719364727386482, −21.12490901238133043008513616853, −20.22909134686616113124625590527, −19.43738040705428181632763888093, −18.65142058623832371868054449250, −18.070305074579312028592992216781, −17.60965238246815973854114672635, −15.81842906375458046189748815507, −15.519166238518024181424157307606, −14.309624996562033432238382508301, −13.887037149213234400554576926721, −12.75201073935117717045775237514, −12.14085082827763701056654660647, −11.256972634158761981893511086370, −10.62343380644811364477787425094, −9.56128776086451659745344758378, −9.20290168090840351854148647742, −7.8654276952041515215307269136, −6.876461995492394437348283059376, −5.76719551911938900729441225252, −5.09439610262454182398082041319, −3.76613387537880917406549837495, −2.98907336356659087695341368226, −2.31878866931387107983905109882, −1.10974575792779614752130629966, 0.47178526803672432959654446723, 1.170998861396494578241576177021, 3.20250074072903611940359690751, 3.99338701804550357539401218496, 4.782398691336668767367636322124, 5.950979608955728289254077326390, 6.31124865991202245527627551543, 7.73488264716136981026951075620, 8.17332771241136031903816076090, 9.06823686074439725968609599391, 9.9206470273446522492531155283, 10.949952018635803802559735977537, 12.15796946019673181506655279314, 12.99992627498891213993977743214, 13.58944202057980944798109347647, 14.16699597698767385871970933924, 15.34894433540871400086464715070, 16.25097494073561553481819633750, 16.539682725358069416951353831, 17.2241137401125217634871576122, 18.25919408985493258408983194742, 19.02814066479645914538308663511, 20.01445369167888399924398843906, 20.92657660882742626436828686686, 21.466260555329446744165126433328

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