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Properties

Label 1-123-123.56-r0-0-0
Degree $1$
Conductor $123$
Sign $0.871 - 0.489i$
Analytic cond. $0.571209$
Root an. cond. $0.571209$
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  + (−0.951 − 0.309i)2-s + (0.809 + 0.587i)4-s + (0.587 − 0.809i)5-s + (0.891 + 0.453i)7-s + (−0.587 − 0.809i)8-s + (−0.809 + 0.587i)10-s + (−0.156 + 0.987i)11-s + (−0.453 − 0.891i)13-s + (−0.707 − 0.707i)14-s + (0.309 + 0.951i)16-s + (0.987 + 0.156i)17-s + (−0.453 + 0.891i)19-s + (0.951 − 0.309i)20-s + (0.453 − 0.891i)22-s + (0.309 − 0.951i)23-s + ⋯
L(s)  = 1  + (−0.951 − 0.309i)2-s + (0.809 + 0.587i)4-s + (0.587 − 0.809i)5-s + (0.891 + 0.453i)7-s + (−0.587 − 0.809i)8-s + (−0.809 + 0.587i)10-s + (−0.156 + 0.987i)11-s + (−0.453 − 0.891i)13-s + (−0.707 − 0.707i)14-s + (0.309 + 0.951i)16-s + (0.987 + 0.156i)17-s + (−0.453 + 0.891i)19-s + (0.951 − 0.309i)20-s + (0.453 − 0.891i)22-s + (0.309 − 0.951i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.871 - 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.871 - 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(123\)    =    \(3 \cdot 41\)
Sign: $0.871 - 0.489i$
Analytic conductor: \(0.571209\)
Root analytic conductor: \(0.571209\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{123} (56, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 123,\ (0:\ ),\ 0.871 - 0.489i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8079075721 - 0.2113009485i\)
\(L(\frac12)\) \(\approx\) \(0.8079075721 - 0.2113009485i\)
\(L(1)\) \(\approx\) \(0.8216771610 - 0.1490487795i\)
\(L(1)\) \(\approx\) \(0.8216771610 - 0.1490487795i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.951 - 0.309i)T \)
5 \( 1 + (0.587 - 0.809i)T \)
7 \( 1 + (0.891 + 0.453i)T \)
11 \( 1 + (-0.156 + 0.987i)T \)
13 \( 1 + (-0.453 - 0.891i)T \)
17 \( 1 + (0.987 + 0.156i)T \)
19 \( 1 + (-0.453 + 0.891i)T \)
23 \( 1 + (0.309 - 0.951i)T \)
29 \( 1 + (0.987 - 0.156i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + (0.951 + 0.309i)T \)
47 \( 1 + (-0.891 + 0.453i)T \)
53 \( 1 + (-0.987 + 0.156i)T \)
59 \( 1 + (-0.309 + 0.951i)T \)
61 \( 1 + (-0.951 + 0.309i)T \)
67 \( 1 + (-0.156 - 0.987i)T \)
71 \( 1 + (0.156 - 0.987i)T \)
73 \( 1 + iT \)
79 \( 1 + (-0.707 + 0.707i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.891 - 0.453i)T \)
97 \( 1 + (0.156 + 0.987i)T \)
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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

−29.17039854901491482459650742616, −27.80752064731895057555081088405, −26.90057382968669059908268812260, −26.24362320944643434216202860047, −25.23043412450533770633169906190, −24.149287448404750987382158473850, −23.35455010565748446741706895261, −21.60094409485237919844351534871, −21.029413858752383847012097188405, −19.47091464982605866473314349035, −18.77006376379534297802805791900, −17.63144750070532226642091868933, −17.009429059543650626890064676268, −15.70694876290574869103396845287, −14.44968468266063764724197112534, −13.84017596307292916694007853277, −11.68437482330378383348470114973, −10.86438518776850336149146056620, −9.888393067009617917252320526326, −8.64231975290860993673362279551, −7.44340989808018028025098880650, −6.46773023084119100048116987045, −5.13175846157601725764516742880, −2.94997600501024705908169492874, −1.48492914685476303049567390958, 1.366327598722212542663463872267, 2.5509249444907247115847771320, 4.633098141830570252565261106471, 5.9722739496193022354598580176, 7.70554701782348051814620987919, 8.473178067941038286857927576336, 9.72315445049755937780774645580, 10.5425973306339623470700708621, 12.17253866285913582595078106056, 12.61190124388174976420326647163, 14.45581971817141335486016322627, 15.58767848670961280424497771899, 16.907463011915977948735996712527, 17.56596113749830875109572642807, 18.45533994726246570230286094995, 19.7500125399802352164895973015, 20.908537153165199190834279004838, 21.119405890497430046217816354891, 22.73474633500566743790720792084, 24.32876801179471394082848194442, 25.02597069135852488875476956018, 25.76909214781755827305227067558, 27.23721511383790438837817008450, 27.88191546228474255612494687352, 28.64918321625284070478824036647

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