Properties

Label 1-3381-3381.2207-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.325 + 0.945i$
Analytic cond. $15.7012$
Root an. cond. $15.7012$
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  + (−0.826 − 0.563i)2-s + (0.365 + 0.930i)4-s + (0.955 − 0.294i)5-s + (0.222 − 0.974i)8-s + (−0.955 − 0.294i)10-s + (0.0747 − 0.997i)11-s + (−0.900 + 0.433i)13-s + (−0.733 + 0.680i)16-s + (−0.988 + 0.149i)17-s + (0.5 + 0.866i)19-s + (0.623 + 0.781i)20-s + (−0.623 + 0.781i)22-s + (0.826 − 0.563i)25-s + (0.988 + 0.149i)26-s + (−0.623 − 0.781i)29-s + ⋯
L(s)  = 1  + (−0.826 − 0.563i)2-s + (0.365 + 0.930i)4-s + (0.955 − 0.294i)5-s + (0.222 − 0.974i)8-s + (−0.955 − 0.294i)10-s + (0.0747 − 0.997i)11-s + (−0.900 + 0.433i)13-s + (−0.733 + 0.680i)16-s + (−0.988 + 0.149i)17-s + (0.5 + 0.866i)19-s + (0.623 + 0.781i)20-s + (−0.623 + 0.781i)22-s + (0.826 − 0.563i)25-s + (0.988 + 0.149i)26-s + (−0.623 − 0.781i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.325 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.325 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.325 + 0.945i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (2207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.325 + 0.945i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5311338825 + 0.3789968541i\)
\(L(\frac12)\) \(\approx\) \(0.5311338825 + 0.3789968541i\)
\(L(1)\) \(\approx\) \(0.7124731069 - 0.1218466710i\)
\(L(1)\) \(\approx\) \(0.7124731069 - 0.1218466710i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.826 - 0.563i)T \)
5 \( 1 + (0.955 - 0.294i)T \)
11 \( 1 + (0.0747 - 0.997i)T \)
13 \( 1 + (-0.900 + 0.433i)T \)
17 \( 1 + (-0.988 + 0.149i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.623 - 0.781i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.365 + 0.930i)T \)
41 \( 1 + (0.222 - 0.974i)T \)
43 \( 1 + (0.222 + 0.974i)T \)
47 \( 1 + (-0.826 - 0.563i)T \)
53 \( 1 + (0.365 + 0.930i)T \)
59 \( 1 + (-0.955 - 0.294i)T \)
61 \( 1 + (-0.365 + 0.930i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.623 + 0.781i)T \)
73 \( 1 + (0.826 - 0.563i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.900 - 0.433i)T \)
89 \( 1 + (0.0747 + 0.997i)T \)
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

−18.347383579564634383457760406678, −17.90057070682529676895313955020, −17.435479892874376804011871162840, −16.83618440157807656714257202946, −15.97567730098515241012732071257, −15.16065039448997465858654280787, −14.76536101240397109759462572655, −14.0179905142271976441776554756, −13.20238249346209518930352057535, −12.50886774374706329070133372632, −11.380197729563182015484183939191, −10.81670245256034824744504850456, −10.009997350905903290043778179672, −9.43256609909556540696837341559, −9.04607069269824528215015099023, −7.93139915698883418620431926400, −7.084558463231587222046563460889, −6.85453722755850048240962386225, −5.80005885228323598548001357859, −5.17088007992427833954470384673, −4.47265974882924181728991303091, −2.96159808526540546018753033466, −2.18964759767947600235420270935, −1.6208413810550688908320980098, −0.252635368982211754421763519, 1.06466550993486706956215799758, 1.846947931379913216223556063486, 2.553732357179811663540154424, 3.41317454220595841348507613954, 4.34496308650542767148991704417, 5.297428789697639643842023681735, 6.18506256469675659415000115862, 6.84992977059831860887654788406, 7.7737097014749690441912226798, 8.53858873690649047009666149787, 9.19572930672575167023812774077, 9.70811535183353108853818813226, 10.48789644757628390426939228593, 11.09860402571970536424272818473, 11.95021327477632536871680517646, 12.52386776601641903134185786327, 13.41025815475187647256093600299, 13.85421490080131764382867294719, 14.768976043814984441146276737575, 15.79059251418510262294504660702, 16.500342643759607607708340723519, 16.9940607144554583448927962548, 17.5605964310261371904337700939, 18.35517292126324875869981705450, 18.83185018796609629471691960984

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