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Properties

Label 1-5519-5519.4762-r0-0-0
Degree $1$
Conductor $5519$
Sign $0.730 - 0.682i$
Analytic cond. $25.6301$
Root an. cond. $25.6301$
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  + (−0.181 − 0.983i)2-s + (−0.633 + 0.773i)3-s + (−0.933 + 0.357i)4-s + (−0.898 − 0.438i)5-s + (0.875 + 0.482i)6-s + (−0.382 − 0.924i)7-s + (0.521 + 0.853i)8-s + (−0.197 − 0.980i)9-s + (−0.268 + 0.963i)10-s + (−0.888 − 0.459i)11-s + (0.315 − 0.949i)12-s + (0.0745 + 0.997i)13-s + (−0.839 + 0.543i)14-s + (0.908 − 0.417i)15-s + (0.744 − 0.667i)16-s + (−0.324 − 0.945i)17-s + ⋯
L(s)  = 1  + (−0.181 − 0.983i)2-s + (−0.633 + 0.773i)3-s + (−0.933 + 0.357i)4-s + (−0.898 − 0.438i)5-s + (0.875 + 0.482i)6-s + (−0.382 − 0.924i)7-s + (0.521 + 0.853i)8-s + (−0.197 − 0.980i)9-s + (−0.268 + 0.963i)10-s + (−0.888 − 0.459i)11-s + (0.315 − 0.949i)12-s + (0.0745 + 0.997i)13-s + (−0.839 + 0.543i)14-s + (0.908 − 0.417i)15-s + (0.744 − 0.667i)16-s + (−0.324 − 0.945i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5519 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.730 - 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5519 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.730 - 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5519\)
Sign: $0.730 - 0.682i$
Analytic conductor: \(25.6301\)
Root analytic conductor: \(25.6301\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5519} (4762, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5519,\ (0:\ ),\ 0.730 - 0.682i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4187344096 - 0.1651827833i\)
\(L(\frac12)\) \(\approx\) \(0.4187344096 - 0.1651827833i\)
\(L(1)\) \(\approx\) \(0.4601438461 - 0.2121346734i\)
\(L(1)\) \(\approx\) \(0.4601438461 - 0.2121346734i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5519 \( 1 \)
good2 \( 1 + (-0.181 - 0.983i)T \)
3 \( 1 + (-0.633 + 0.773i)T \)
5 \( 1 + (-0.898 - 0.438i)T \)
7 \( 1 + (-0.382 - 0.924i)T \)
11 \( 1 + (-0.888 - 0.459i)T \)
13 \( 1 + (0.0745 + 0.997i)T \)
17 \( 1 + (-0.324 - 0.945i)T \)
19 \( 1 + (-0.0960 - 0.995i)T \)
23 \( 1 + (0.398 + 0.917i)T \)
29 \( 1 + (-0.365 - 0.930i)T \)
31 \( 1 + (0.943 - 0.330i)T \)
37 \( 1 + (-0.177 - 0.984i)T \)
41 \( 1 + (0.101 + 0.994i)T \)
43 \( 1 + (0.750 - 0.660i)T \)
47 \( 1 + (0.935 + 0.354i)T \)
53 \( 1 + (-0.228 + 0.973i)T \)
59 \( 1 + (-0.303 + 0.952i)T \)
61 \( 1 + (-0.432 - 0.901i)T \)
67 \( 1 + (0.368 + 0.929i)T \)
71 \( 1 + (-0.865 + 0.501i)T \)
73 \( 1 + (-0.996 - 0.0875i)T \)
79 \( 1 + (-0.826 - 0.562i)T \)
83 \( 1 + (-0.883 - 0.469i)T \)
89 \( 1 + (-0.496 + 0.867i)T \)
97 \( 1 + (0.312 + 0.949i)T \)
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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

−18.04564635307668500104554490785, −17.31610946213687264511352862421, −16.61507487599826654577933385926, −15.837984625281735465023473625624, −15.49047791179756192752085926872, −14.843249212534745435956154774713, −14.17341977399035204540830879982, −13.12448913585724463390299149274, −12.57400365702774989736041685075, −12.37422329138376977891392940832, −11.272864246151331675433092643009, −10.351981506583202017785880103804, −10.238622564410899348283675387688, −8.732220989221834234741781858559, −8.35563545182599220354712722851, −7.726029462918533039941334095211, −7.115888534052062588272125070553, −6.358377763415539907768030330244, −5.861312239465920360558868630821, −5.13733471468365613306435935373, −4.45261897485373374774551740748, −3.35749423773403889305015087993, −2.57479616641913552455854101285, −1.494959575108800171709155940504, −0.32156140798677299110229775022, 0.48836516784417105236581855265, 1.14120146005133678545399834725, 2.61105912619398213792924515312, 3.20952683337750817062084409860, 4.21681129934938813641552319971, 4.314629625059517567987433567798, 5.10888661561924266209466422190, 5.9826624039335274955148213690, 7.1851156462834443911101535590, 7.59362358825098547162008482429, 8.71804048559085673139414265049, 9.2045049570243889602277899320, 9.80957458102777428370349038734, 10.63928415078689659359377115858, 11.20196030637618583114127331717, 11.53226271933117600304748964499, 12.23590556500325795990355250574, 13.150610236944093813850382726579, 13.544961229391005891479200710084, 14.348858451723533619305734962999, 15.48776902655185267727810871594, 15.873768269886030813121431036896, 16.52408215361775047394361142585, 17.1983200795318886684694741528, 17.627409668292370210129098631253

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