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Properties

Label 1-621-621.482-r0-0-0
Degree $1$
Conductor $621$
Sign $0.835 + 0.549i$
Analytic cond. $2.88391$
Root an. cond. $2.88391$
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  + (−0.766 − 0.642i)2-s + (0.173 + 0.984i)4-s + (−0.939 − 0.342i)5-s + (−0.173 + 0.984i)7-s + (0.5 − 0.866i)8-s + (0.5 + 0.866i)10-s + (−0.939 + 0.342i)11-s + (0.766 − 0.642i)13-s + (0.766 − 0.642i)14-s + (−0.939 + 0.342i)16-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + (0.173 − 0.984i)20-s + (0.939 + 0.342i)22-s + (0.766 + 0.642i)25-s − 26-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)2-s + (0.173 + 0.984i)4-s + (−0.939 − 0.342i)5-s + (−0.173 + 0.984i)7-s + (0.5 − 0.866i)8-s + (0.5 + 0.866i)10-s + (−0.939 + 0.342i)11-s + (0.766 − 0.642i)13-s + (0.766 − 0.642i)14-s + (−0.939 + 0.342i)16-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + (0.173 − 0.984i)20-s + (0.939 + 0.342i)22-s + (0.766 + 0.642i)25-s − 26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 621 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 621 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(621\)    =    \(3^{3} \cdot 23\)
Sign: $0.835 + 0.549i$
Analytic conductor: \(2.88391\)
Root analytic conductor: \(2.88391\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{621} (482, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 621,\ (0:\ ),\ 0.835 + 0.549i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5265000067 + 0.1576237547i\)
\(L(\frac12)\) \(\approx\) \(0.5265000067 + 0.1576237547i\)
\(L(1)\) \(\approx\) \(0.5733637128 - 0.07023904637i\)
\(L(1)\) \(\approx\) \(0.5733637128 - 0.07023904637i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.766 - 0.642i)T \)
5 \( 1 + (-0.939 - 0.342i)T \)
7 \( 1 + (-0.173 + 0.984i)T \)
11 \( 1 + (-0.939 + 0.342i)T \)
13 \( 1 + (0.766 - 0.642i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.766 - 0.642i)T \)
31 \( 1 + (0.173 + 0.984i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (-0.766 + 0.642i)T \)
43 \( 1 + (0.939 - 0.342i)T \)
47 \( 1 + (-0.173 + 0.984i)T \)
53 \( 1 + T \)
59 \( 1 + (0.939 + 0.342i)T \)
61 \( 1 + (-0.173 + 0.984i)T \)
67 \( 1 + (-0.766 + 0.642i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (0.766 + 0.642i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.939 - 0.342i)T \)
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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

−23.223333274379730872148370809282, −22.45470084368604381901148453287, −20.99947145810557847062605260656, −20.22845981369761910463495530061, −19.456974190251903333332257722241, −18.69181562232589975233651750494, −18.110956663830381990221665854422, −16.89865923139926386006983865058, −16.31560890384097627809446336572, −15.631559902518712197991651633093, −14.75177745581360520421849086154, −13.8847838893299974644429336189, −12.9811790033141298632588025805, −11.547632248938483722193337485, −10.81143025696596215587006494591, −10.25786382992186391783103226109, −9.02160595670473488354730512077, −8.07437191627905862721634591466, −7.502509273077260047470951714146, −6.623854533315100854759496839622, −5.668850871080593472641562285002, −4.33256360033472701568325584817, −3.48062072204066969487477179575, −1.873825410778728713978771437055, −0.46052031135107428290406268187, 0.95843414285426232300322402053, 2.508898542655582871361220155292, 3.14769317772251725315033236748, 4.4148810687741906513270013773, 5.428269990891643688354760764443, 6.90884854818017253801617406424, 7.79981983142306228186954547705, 8.56737586588740866514603623953, 9.26064731928920349370426453663, 10.323172127664883803817293116020, 11.345530768383330112978176198621, 11.82292989318299560736562244277, 12.84562700512209567387043095359, 13.364068302961692825555071822154, 15.15689125201809929161165674412, 15.76995802299850446352445692295, 16.24449452633732472195417457251, 17.59599801079735893878804247252, 18.2650069906568548457523206708, 18.87794978893032060787194293172, 19.81104519178759444959171640579, 20.48474277561140723968219866990, 21.13189095237117539582620607656, 22.244745437421547046623202888407, 22.84313642431214039581460097244

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