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Properties

Label 1-212-212.127-r0-0-0
Degree $1$
Conductor $212$
Sign $0.742 - 0.670i$
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  + (−0.663 − 0.748i)3-s + (0.992 + 0.120i)5-s + (0.568 − 0.822i)7-s + (−0.120 + 0.992i)9-s + (0.885 + 0.464i)11-s + (−0.354 + 0.935i)13-s + (−0.568 − 0.822i)15-s + (0.970 − 0.239i)17-s + (−0.935 − 0.354i)19-s + (−0.992 + 0.120i)21-s − i·23-s + (0.970 + 0.239i)25-s + (0.822 − 0.568i)27-s + (−0.885 + 0.464i)29-s + (0.464 + 0.885i)31-s + ⋯
L(s)  = 1  + (−0.663 − 0.748i)3-s + (0.992 + 0.120i)5-s + (0.568 − 0.822i)7-s + (−0.120 + 0.992i)9-s + (0.885 + 0.464i)11-s + (−0.354 + 0.935i)13-s + (−0.568 − 0.822i)15-s + (0.970 − 0.239i)17-s + (−0.935 − 0.354i)19-s + (−0.992 + 0.120i)21-s − i·23-s + (0.970 + 0.239i)25-s + (0.822 − 0.568i)27-s + (−0.885 + 0.464i)29-s + (0.464 + 0.885i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 212 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.742 - 0.670i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.742 - 0.670i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(212\)    =    \(2^{2} \cdot 53\)
Sign: $0.742 - 0.670i$
Analytic conductor: \(0.984523\)
Root analytic conductor: \(0.984523\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{212} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 212,\ (0:\ ),\ 0.742 - 0.670i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.137588533 - 0.4375725115i\)
\(L(\frac12)\) \(\approx\) \(1.137588533 - 0.4375725115i\)
\(L(1)\) \(\approx\) \(1.064289458 - 0.2569197200i\)
\(L(1)\) \(\approx\) \(1.064289458 - 0.2569197200i\)

Euler product

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

−27.05149524066736696348168377834, −25.70945631934525195329646527968, −25.00694763120362461091980044781, −24.02776351431132248020413504146, −22.77647673094629646618607478602, −21.92835413161081366375059415406, −21.34442173924444225535207791604, −20.55187624217205262211327326682, −19.08381956145675510717877166584, −17.94249407295425193831157509379, −17.2126029619576895567142075072, −16.50564355274129930723633024071, −15.0682662910543562484985643454, −14.604139912484372141575524309237, −13.12260214728740806013840804590, −12.032828818973803681871859259436, −11.13325828778679302102638828236, −9.96834610444159806130156665229, −9.274416422871515888203100923494, −8.04798474885241903324113887535, −6.11873679265250364127772025490, −5.720241442522090766060934318179, −4.55241515733441280560022289741, −3.065596456208550439052529239029, −1.444391409120418469845097674519, 1.264496121063984884739264024268, 2.225197786837579997019595178179, 4.27379370685240814992251698544, 5.37861108724403014126870299671, 6.64618924463865897353072464629, 7.188141159731906506799616800625, 8.69956062374322040311132731701, 10.005696088440215041011378961234, 10.91558309877582363421154284179, 11.98916808117205653486378949224, 12.957927779951255486799322893758, 14.07661761530785840393203650804, 14.55556200416549453421878299292, 16.673730718208231940487034426815, 16.97049496428518388575307702245, 17.891900568438088222584045888677, 18.80209706524750633621657451735, 19.86729352680265611299396392703, 21.03939173230948410686501959131, 21.92994952532448695488112684946, 22.90983826108391412925094844172, 23.769528878895473607473893587029, 24.658075019618973573018240205942, 25.43344252194991813491891605612, 26.50050583145252481268813905485

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