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Properties

Label 1-212-212.39-r0-0-0
Degree $1$
Conductor $212$
Sign $0.876 - 0.480i$
Analytic cond. $0.984523$
Root an. cond. $0.984523$
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  + (0.822 − 0.568i)3-s + (0.935 + 0.354i)5-s + (−0.970 − 0.239i)7-s + (0.354 − 0.935i)9-s + (0.120 + 0.992i)11-s + (0.885 − 0.464i)13-s + (0.970 − 0.239i)15-s + (0.748 − 0.663i)17-s + (−0.464 − 0.885i)19-s + (−0.935 + 0.354i)21-s − i·23-s + (0.748 + 0.663i)25-s + (−0.239 − 0.970i)27-s + (−0.120 + 0.992i)29-s + (−0.992 − 0.120i)31-s + ⋯
L(s)  = 1  + (0.822 − 0.568i)3-s + (0.935 + 0.354i)5-s + (−0.970 − 0.239i)7-s + (0.354 − 0.935i)9-s + (0.120 + 0.992i)11-s + (0.885 − 0.464i)13-s + (0.970 − 0.239i)15-s + (0.748 − 0.663i)17-s + (−0.464 − 0.885i)19-s + (−0.935 + 0.354i)21-s − i·23-s + (0.748 + 0.663i)25-s + (−0.239 − 0.970i)27-s + (−0.120 + 0.992i)29-s + (−0.992 − 0.120i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 212 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(212\)    =    \(2^{2} \cdot 53\)
Sign: $0.876 - 0.480i$
Analytic conductor: \(0.984523\)
Root analytic conductor: \(0.984523\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{212} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 212,\ (0:\ ),\ 0.876 - 0.480i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.620825394 - 0.4149896336i\)
\(L(\frac12)\) \(\approx\) \(1.620825394 - 0.4149896336i\)
\(L(1)\) \(\approx\) \(1.429994567 - 0.2328919572i\)
\(L(1)\) \(\approx\) \(1.429994567 - 0.2328919572i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
53 \( 1 \)
good3 \( 1 + (0.822 - 0.568i)T \)
5 \( 1 + (0.935 + 0.354i)T \)
7 \( 1 + (-0.970 - 0.239i)T \)
11 \( 1 + (0.120 + 0.992i)T \)
13 \( 1 + (0.885 - 0.464i)T \)
17 \( 1 + (0.748 - 0.663i)T \)
19 \( 1 + (-0.464 - 0.885i)T \)
23 \( 1 - iT \)
29 \( 1 + (-0.120 + 0.992i)T \)
31 \( 1 + (-0.992 - 0.120i)T \)
37 \( 1 + (-0.568 - 0.822i)T \)
41 \( 1 + (-0.992 + 0.120i)T \)
43 \( 1 + (0.568 - 0.822i)T \)
47 \( 1 + (0.354 + 0.935i)T \)
59 \( 1 + (-0.354 - 0.935i)T \)
61 \( 1 + (-0.663 + 0.748i)T \)
67 \( 1 + (-0.464 + 0.885i)T \)
71 \( 1 + (0.822 + 0.568i)T \)
73 \( 1 + (-0.663 - 0.748i)T \)
79 \( 1 + (0.239 + 0.970i)T \)
83 \( 1 - iT \)
89 \( 1 + (-0.748 + 0.663i)T \)
97 \( 1 + (-0.354 + 0.935i)T \)
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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

−26.47358425780992710180414741196, −25.78595534931986081090728288390, −25.15629513645926314195640661338, −24.18852960403133916473549552181, −22.807430720867172852749005675264, −21.741809241942057444954945984820, −21.19063035992837863980153637175, −20.30969572962365574843725961052, −19.07983253051346052381982397106, −18.5676499411303259237970205052, −16.67079160519943916817745402545, −16.49772905751789330741159471470, −15.186602901570190264501822184534, −14.06685139952699574107268942811, −13.41757717098058631111062745220, −12.4202866558592670487528628038, −10.71829913706599483672546671487, −9.915078803612908877935936006954, −8.94535751178783933593326641910, −8.26084074384236647887506365346, −6.433710164187265757957947480537, −5.62177690387598039501688231261, −4.02653633750542492645514035919, −3.05548501006783925323448274163, −1.70202923653984166286982740238, 1.456883269318582837970201402591, 2.71683376060044562132178061811, 3.67881767631680959637087879806, 5.53816352284229735087207274487, 6.7404637552820170526774838394, 7.39374742805557033611307865283, 9.00815267842569249817124415921, 9.619137897382006687287797338754, 10.70262204369337569836596180319, 12.382095079554822011647193529272, 13.1683306889484626739208689948, 13.87853912765080104866369080855, 14.90767977088613046589649926622, 15.91569362603140277396176221634, 17.32150200238646717760814813092, 18.12969102163247063361071456045, 18.98698663397254439618659375289, 20.03950691591362929610572685245, 20.73745688268513356359426431442, 21.88161293240307128199434412693, 22.92546681629170525951720259193, 23.76320027810166362260870397183, 25.22658608353164569038720883406, 25.5815063000620558267974130250, 26.08243565956586892849750804152

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