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Properties

Label 1-4719-4719.425-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: $\chi_{4719} (425, \cdot )$
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

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

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