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Properties

Label 1-212-212.23-r0-0-0
Degree $1$
Conductor $212$
Sign $-0.138 + 0.990i$
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  + i·3-s + i·5-s + 7-s − 9-s + 11-s + 13-s − 15-s − 17-s + i·19-s + i·21-s − i·23-s − 25-s − i·27-s − 29-s + i·31-s + ⋯
L(s)  = 1  + i·3-s + i·5-s + 7-s − 9-s + 11-s + 13-s − 15-s − 17-s + i·19-s + i·21-s − i·23-s − 25-s − i·27-s − 29-s + i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 212 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.138 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.138 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(212\)    =    \(2^{2} \cdot 53\)
Sign: $-0.138 + 0.990i$
Analytic conductor: \(0.984523\)
Root analytic conductor: \(0.984523\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{212} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 212,\ (0:\ ),\ -0.138 + 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8448251359 + 0.9713926212i\)
\(L(\frac12)\) \(\approx\) \(0.8448251359 + 0.9713926212i\)
\(L(1)\) \(\approx\) \(1.009347814 + 0.5689160498i\)
\(L(1)\) \(\approx\) \(1.009347814 + 0.5689160498i\)

Euler product

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

−26.24440523113699102690545472974, −25.213481880476687417510974104462, −24.32873437825071067557590307580, −24.019206406687277775196533025289, −22.88078250839330301173816816595, −21.65360723940769714301324567636, −20.50474258448712438199073501644, −19.90642815223404830028566707296, −18.83446159582304399997041336189, −17.56299178029596061603546384671, −17.33060660397642038721771456670, −15.94277184491901976568227267123, −14.71318312365068750273968653640, −13.56750827113550111614740604447, −13.01572160382183754253416306989, −11.613551920764422165737901241404, −11.29901497654080206710424522286, −9.19354565660599017808701669180, −8.55831242503491924893741836316, −7.516740334432643231734276036897, −6.2964251946882644876337388035, −5.17681628749719901206217017389, −3.92590052472514117386300479323, −2.00423086915382249665877975856, −1.09727618539121721622667746778, 1.93647544291610846710389048898, 3.48612135192018261300865347359, 4.301909071238966083926416891436, 5.69810221865560018975099411749, 6.77153576390151403376769795882, 8.26015889210504992999248828890, 9.13943886840975892285295722656, 10.531335393308183139388906663973, 11.01585747838710850207875182878, 11.98232438165130628386071578877, 13.861650656139061614443875713500, 14.50134325202706425342600430284, 15.26229611895336337129261035395, 16.33380263378483429610294650483, 17.43810735878492341467577257018, 18.2426895717604638643055345422, 19.427926391183181879637276368170, 20.58806968213733634904871864292, 21.22859672934665463044543132316, 22.39229642482400597152250502636, 22.74720942999961754081587515608, 24.11455901968477301760773262712, 25.24317796651072781308998636805, 26.1686446876893589764695311076, 27.026420140053125136220421820573

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