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Properties

Label 1-4019-4019.113-r0-0-0
Degree $1$
Conductor $4019$
Sign $0.435 + 0.900i$
Analytic cond. $18.6641$
Root an. cond. $18.6641$
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  + (−0.662 + 0.748i)2-s + (−0.997 + 0.0640i)3-s + (−0.120 − 0.992i)4-s + (0.444 − 0.895i)5-s + (0.613 − 0.789i)6-s + (0.986 + 0.165i)7-s + (0.823 + 0.567i)8-s + (0.991 − 0.127i)9-s + (0.375 + 0.926i)10-s + (−0.627 + 0.778i)11-s + (0.184 + 0.982i)12-s + (0.320 + 0.947i)13-s + (−0.777 + 0.628i)14-s + (−0.386 + 0.922i)15-s + (−0.970 + 0.240i)16-s + (0.970 + 0.241i)17-s + ⋯
L(s)  = 1  + (−0.662 + 0.748i)2-s + (−0.997 + 0.0640i)3-s + (−0.120 − 0.992i)4-s + (0.444 − 0.895i)5-s + (0.613 − 0.789i)6-s + (0.986 + 0.165i)7-s + (0.823 + 0.567i)8-s + (0.991 − 0.127i)9-s + (0.375 + 0.926i)10-s + (−0.627 + 0.778i)11-s + (0.184 + 0.982i)12-s + (0.320 + 0.947i)13-s + (−0.777 + 0.628i)14-s + (−0.386 + 0.922i)15-s + (−0.970 + 0.240i)16-s + (0.970 + 0.241i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4019 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.435 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4019 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.435 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4019\)
Sign: $0.435 + 0.900i$
Analytic conductor: \(18.6641\)
Root analytic conductor: \(18.6641\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4019} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4019,\ (0:\ ),\ 0.435 + 0.900i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.025014264 + 0.6424979121i\)
\(L(\frac12)\) \(\approx\) \(1.025014264 + 0.6424979121i\)
\(L(1)\) \(\approx\) \(0.7331379316 + 0.2424110746i\)
\(L(1)\) \(\approx\) \(0.7331379316 + 0.2424110746i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4019 \( 1 \)
good2 \( 1 + (-0.662 + 0.748i)T \)
3 \( 1 + (-0.997 + 0.0640i)T \)
5 \( 1 + (0.444 - 0.895i)T \)
7 \( 1 + (0.986 + 0.165i)T \)
11 \( 1 + (-0.627 + 0.778i)T \)
13 \( 1 + (0.320 + 0.947i)T \)
17 \( 1 + (0.970 + 0.241i)T \)
19 \( 1 + (-0.289 + 0.957i)T \)
23 \( 1 + (0.413 + 0.910i)T \)
29 \( 1 + (0.999 + 0.00938i)T \)
31 \( 1 + (0.983 - 0.180i)T \)
37 \( 1 + (0.993 - 0.112i)T \)
41 \( 1 + (-0.870 - 0.492i)T \)
43 \( 1 + (0.833 + 0.552i)T \)
47 \( 1 + (-0.164 - 0.986i)T \)
53 \( 1 + (0.955 - 0.295i)T \)
59 \( 1 + (-0.0429 + 0.999i)T \)
61 \( 1 + (0.387 - 0.921i)T \)
67 \( 1 + (0.962 - 0.271i)T \)
71 \( 1 + (0.983 - 0.183i)T \)
73 \( 1 + (-0.951 - 0.306i)T \)
79 \( 1 + (-0.535 - 0.844i)T \)
83 \( 1 + (0.803 - 0.595i)T \)
89 \( 1 + (-0.848 - 0.529i)T \)
97 \( 1 + (-0.306 + 0.951i)T \)
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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

−18.42910217505525825937429149971, −17.65500171601932773469551046616, −17.38891051539187232536788007275, −16.58111405036969503625723813887, −15.806422993163672400671351778182, −15.084507887212608207705861541500, −14.04417645507208535432255385013, −13.44709607901465593001781107383, −12.72542736728732601874764500544, −11.93776027328268100092502664906, −11.155481481837939938905191153745, −10.88389543777900404994248031308, −10.286870951893577488734346970248, −9.73069731767745124284088439538, −8.48049733073159351250279922307, −8.00408065092482029810957619322, −7.19004561409756045037396287076, −6.493741317344075815619438723754, −5.571913642337590283743657301978, −4.90157048344966420569671019140, −4.01230177423283522455425908908, −2.880468121292950022868852525724, −2.50723360452448168328394563408, −1.1957049861013711148037295064, −0.72329383761712593693696976023, 0.8883512933569310550381923783, 1.516665185849566057044187600342, 2.138993772370468002674566816761, 4.08105530756705000254458574313, 4.69440049648851449853074929613, 5.29896473491766599448150535600, 5.82061663987427364937424469139, 6.58566634430450992650613504299, 7.49332549857612632978727854479, 8.08118568735367696854400524763, 8.78753805457144966265196192728, 9.742358060118004894084038626154, 10.08318802336371390039078669434, 10.888611496121483815125309177521, 11.77864302798112540695480167776, 12.21451424017060227979106922997, 13.20217855205260210671306233824, 13.89205694391138135865307568155, 14.71379960323864041888721625477, 15.401245518507263440232274446833, 16.12504754354120669874491918433, 16.658622001732506286936899599, 17.28778398727100082857763811568, 17.65604638638794624035158486526, 18.41108333801960348406923338646

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