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Properties

Label 1-4719-4719.4397-r0-0-0
Degree $1$
Conductor $4719$
Sign $0.290 + 0.956i$
Analytic cond. $21.9149$
Root an. cond. $21.9149$
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  + (0.964 + 0.263i)2-s + (0.861 + 0.508i)4-s + (0.985 − 0.170i)5-s + (0.683 + 0.730i)7-s + (0.696 + 0.717i)8-s + (0.995 + 0.0950i)10-s + (0.466 + 0.884i)14-s + (0.483 + 0.875i)16-s + (−0.851 + 0.524i)17-s + (0.991 + 0.132i)19-s + (0.935 + 0.353i)20-s + (−0.981 + 0.189i)23-s + (0.941 − 0.336i)25-s + (0.217 + 0.976i)28-s + (−0.761 + 0.647i)29-s + ⋯
L(s)  = 1  + (0.964 + 0.263i)2-s + (0.861 + 0.508i)4-s + (0.985 − 0.170i)5-s + (0.683 + 0.730i)7-s + (0.696 + 0.717i)8-s + (0.995 + 0.0950i)10-s + (0.466 + 0.884i)14-s + (0.483 + 0.875i)16-s + (−0.851 + 0.524i)17-s + (0.991 + 0.132i)19-s + (0.935 + 0.353i)20-s + (−0.981 + 0.189i)23-s + (0.941 − 0.336i)25-s + (0.217 + 0.976i)28-s + (−0.761 + 0.647i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4719 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4719 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4719\)    =    \(3 \cdot 11^{2} \cdot 13\)
Sign: $0.290 + 0.956i$
Analytic conductor: \(21.9149\)
Root analytic conductor: \(21.9149\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4719} (4397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4719,\ (0:\ ),\ 0.290 + 0.956i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.962060459 + 2.939052069i\)
\(L(\frac12)\) \(\approx\) \(3.962060459 + 2.939052069i\)
\(L(1)\) \(\approx\) \(2.357954028 + 0.8294250365i\)
\(L(1)\) \(\approx\) \(2.357954028 + 0.8294250365i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.964 + 0.263i)T \)
5 \( 1 + (0.985 - 0.170i)T \)
7 \( 1 + (0.683 + 0.730i)T \)
17 \( 1 + (-0.851 + 0.524i)T \)
19 \( 1 + (0.991 + 0.132i)T \)
23 \( 1 + (-0.981 + 0.189i)T \)
29 \( 1 + (-0.761 + 0.647i)T \)
31 \( 1 + (0.993 - 0.113i)T \)
37 \( 1 + (0.749 + 0.662i)T \)
41 \( 1 + (-0.625 - 0.780i)T \)
43 \( 1 + (0.786 - 0.618i)T \)
47 \( 1 + (0.998 - 0.0570i)T \)
53 \( 1 + (-0.516 - 0.856i)T \)
59 \( 1 + (0.625 - 0.780i)T \)
61 \( 1 + (-0.964 + 0.263i)T \)
67 \( 1 + (0.0475 + 0.998i)T \)
71 \( 1 + (0.432 - 0.901i)T \)
73 \( 1 + (-0.0855 - 0.996i)T \)
79 \( 1 + (-0.897 - 0.441i)T \)
83 \( 1 + (-0.921 - 0.389i)T \)
89 \( 1 + (-0.723 + 0.690i)T \)
97 \( 1 + (0.345 + 0.938i)T \)
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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

−18.089487140236370221526754871027, −17.296006224942769137950929544295, −16.701523540412301202392260604859, −15.86189958118866971479262557890, −15.245789835163589122211815210772, −14.31828563683217353713922264402, −13.98862313354148280115120863185, −13.491828730770362940310914848087, −12.83261724078626559116184111622, −11.93647463675352816139716029745, −11.23648715916880756665460763474, −10.79609026567575858878349771395, −9.8820856259677469776912004799, −9.54199532687521987089832640879, −8.33842630418503977679495481084, −7.42292541288462611728308602837, −6.91817078042559095677053438102, −5.990545604712700742707572254419, −5.56088147386500762153992394999, −4.53685198954967152933779357339, −4.25365925179846193721505946218, −3.06718317850725218336345941257, −2.43790761760213830273713399881, −1.65449524650870709265566703888, −0.88348277393753080166341605953, 1.33868702665373457700990397246, 2.038289701016592053655261175136, 2.59651820427925777451615005629, 3.57412352211957238683437911918, 4.48040124553700286404516517673, 5.13656368663874426097856411521, 5.757655477597608484398543148008, 6.25007477681750813061085527851, 7.11830331557027412133362092689, 7.96072667666560748971039340043, 8.63241620034750957832984507573, 9.380735049039698422989359740676, 10.2870063822095471404645758224, 10.98043083850275223107552866293, 11.77738207074133493371689394157, 12.2586882142613954883745676373, 13.07931971078542113643434226985, 13.64831166513666123309921628503, 14.22956629491126527070577978296, 14.8282903270326218462456433130, 15.57437427496732763627977661480, 16.07843008671536962370097846579, 17.02401078728619669851508251008, 17.493466350950922433926881812924, 18.12743859705065204004677338328

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