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Properties

Label 1-212-212.3-r0-0-0
Degree $1$
Conductor $212$
Sign $0.370 + 0.929i$
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  + (0.935 + 0.354i)3-s + (−0.663 + 0.748i)5-s + (0.885 + 0.464i)7-s + (0.748 + 0.663i)9-s + (−0.970 + 0.239i)11-s + (0.568 − 0.822i)13-s + (−0.885 + 0.464i)15-s + (−0.120 + 0.992i)17-s + (−0.822 − 0.568i)19-s + (0.663 + 0.748i)21-s − i·23-s + (−0.120 − 0.992i)25-s + (0.464 + 0.885i)27-s + (0.970 + 0.239i)29-s + (−0.239 + 0.970i)31-s + ⋯
L(s)  = 1  + (0.935 + 0.354i)3-s + (−0.663 + 0.748i)5-s + (0.885 + 0.464i)7-s + (0.748 + 0.663i)9-s + (−0.970 + 0.239i)11-s + (0.568 − 0.822i)13-s + (−0.885 + 0.464i)15-s + (−0.120 + 0.992i)17-s + (−0.822 − 0.568i)19-s + (0.663 + 0.748i)21-s − i·23-s + (−0.120 − 0.992i)25-s + (0.464 + 0.885i)27-s + (0.970 + 0.239i)29-s + (−0.239 + 0.970i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 212 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.370 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.370 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(212\)    =    \(2^{2} \cdot 53\)
Sign: $0.370 + 0.929i$
Analytic conductor: \(0.984523\)
Root analytic conductor: \(0.984523\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{212} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 212,\ (0:\ ),\ 0.370 + 0.929i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.235627149 + 0.8378347659i\)
\(L(\frac12)\) \(\approx\) \(1.235627149 + 0.8378347659i\)
\(L(1)\) \(\approx\) \(1.250320370 + 0.4370647737i\)
\(L(1)\) \(\approx\) \(1.250320370 + 0.4370647737i\)

Euler product

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

−26.64302804035418773071435296098, −25.498074311072570111334801698198, −24.501720037477353233260550632760, −23.83356346454474444670175757656, −23.171545904729477687616492357960, −21.287417653692427663828084411245, −20.74855982919475887640847312784, −20.06932477470173308596633083751, −18.87877332663113515258133312846, −18.251011358453697771138408987029, −16.81069538412613528989911597638, −15.898848872162181229432610678799, −14.87475621741451726252629140452, −13.87066102717768628858313619678, −13.07260798329232531657116263949, −11.96768228818063606593936203843, −10.90163543398538200794193995198, −9.489560706064061590245385108626, −8.255490719940567812481481156269, −7.98550578724429249079155978146, −6.625401749885369596706498812097, −4.815208749932649655133646066980, −4.02528606302711912902675432063, −2.51457008635080345233504783135, −1.124980354565590969164004021998, 2.01668334974327974551381267070, 3.09565970047329306532610462390, 4.203212263341068584220253702740, 5.462503935024725292228383790719, 7.155904171910499141609682220723, 8.10580807070121831062687695531, 8.73056646654097162225841750035, 10.43303787636468159812335869596, 10.84179170950264752194087121969, 12.330071537487856817103319929778, 13.45501248800551042570664407493, 14.558589839864096339391351991857, 15.3624630802337778575259391040, 15.74296173246185542151262114189, 17.579868477411465660661233124199, 18.41049217894315252046687356973, 19.36167191202894427365758360321, 20.19262540579816002776393561468, 21.29015030550747883868033080674, 21.834492485131690055684382642887, 23.28693635877925244609749327148, 23.93865487687140484736748186268, 25.29817758827745101291752049392, 25.847886581047349910845382932, 26.880652102686230199268126583127

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