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Properties

Label 1-124-124.87-r1-0-0
Degree $1$
Conductor $124$
Sign $-0.920 - 0.390i$
Analytic cond. $13.3256$
Root an. cond. $13.3256$
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  + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)7-s + (−0.5 − 0.866i)9-s + (0.5 + 0.866i)11-s + (−0.5 − 0.866i)13-s − 15-s + (−0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (−0.5 − 0.866i)21-s − 23-s + (−0.5 + 0.866i)25-s − 27-s + 29-s + 33-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)7-s + (−0.5 − 0.866i)9-s + (0.5 + 0.866i)11-s + (−0.5 − 0.866i)13-s − 15-s + (−0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (−0.5 − 0.866i)21-s − 23-s + (−0.5 + 0.866i)25-s − 27-s + 29-s + 33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.920 - 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.920 - 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(124\)    =    \(2^{2} \cdot 31\)
Sign: $-0.920 - 0.390i$
Analytic conductor: \(13.3256\)
Root analytic conductor: \(13.3256\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{124} (87, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 124,\ (1:\ ),\ -0.920 - 0.390i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3086910007 - 1.518434124i\)
\(L(\frac12)\) \(\approx\) \(0.3086910007 - 1.518434124i\)
\(L(1)\) \(\approx\) \(0.9010504886 - 0.6794154538i\)
\(L(1)\) \(\approx\) \(0.9010504886 - 0.6794154538i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
31 \( 1 \)
good3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 - T \)
29 \( 1 + T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + T \)
97 \( 1 + T \)
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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

−29.06678474242349056987496321373, −27.80993549162088736755886411583, −26.99141959343495360526611059089, −26.43935187497085310683651401331, −25.15914750786188667310311021912, −24.27812571693457874173587186161, −22.753864607913846142349296534994, −21.921702731248955487014084947646, −21.24704067608625315022002552945, −19.87936781791587431913964126517, −19.02509370947656095648875408633, −18.011814356752078651639286977894, −16.38558546014322030950248177454, −15.66287837072141284723678965898, −14.44016226247912859665637661103, −14.06732321452962130327415146858, −11.88986548866306458929797288674, −11.24635275038310370754828658306, −9.916714836636461671194595245996, −8.82099516599886063241277221321, −7.77056166135495686561897683109, −6.1970215408048574890782053181, −4.723120251041703983057662482113, −3.47336544024123318218140286857, −2.30736020264089067479409899093, 0.588182719790064703067288512107, 1.87552117060048554520137683756, 3.7237872869665075578988931911, 4.95854296965368270546306795168, 6.76761550403938826238440221798, 7.77288891592561222395589179692, 8.608637256504189099256100270410, 10.020335332678744021091138022381, 11.635235143143695494528851600262, 12.5479960923198772615159201524, 13.46514649647429685760927849680, 14.6045508932305322666974161370, 15.68054454944815771935661189643, 17.3510558530232488677214755350, 17.63868286252788128781625130716, 19.39433295080344146600845370889, 20.05470846541809915271677238887, 20.62232211291252404841529187623, 22.34445585550389507829481222013, 23.60599469875971222116757520053, 24.13697519062856561906909409895, 25.06852500023259458154058239290, 26.15952296142324803695402864822, 27.27831220232034521424555529451, 28.25238643832065031135026797829

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