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Properties

Label 1-42e2-1764.1039-r0-0-0
Degree $1$
Conductor $1764$
Sign $0.763 - 0.645i$
Analytic cond. $8.19198$
Root an. cond. $8.19198$
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  + (−0.623 − 0.781i)5-s + (0.222 + 0.974i)11-s + (0.733 − 0.680i)13-s + (−0.0747 − 0.997i)17-s + (−0.5 + 0.866i)19-s + (0.900 − 0.433i)23-s + (−0.222 + 0.974i)25-s + (0.826 − 0.563i)29-s + (−0.5 + 0.866i)31-s + (0.826 − 0.563i)37-s + (−0.365 + 0.930i)41-s + (−0.365 − 0.930i)43-s + (−0.733 + 0.680i)47-s + (0.826 + 0.563i)53-s + (0.623 − 0.781i)55-s + ⋯
L(s)  = 1  + (−0.623 − 0.781i)5-s + (0.222 + 0.974i)11-s + (0.733 − 0.680i)13-s + (−0.0747 − 0.997i)17-s + (−0.5 + 0.866i)19-s + (0.900 − 0.433i)23-s + (−0.222 + 0.974i)25-s + (0.826 − 0.563i)29-s + (−0.5 + 0.866i)31-s + (0.826 − 0.563i)37-s + (−0.365 + 0.930i)41-s + (−0.365 − 0.930i)43-s + (−0.733 + 0.680i)47-s + (0.826 + 0.563i)53-s + (0.623 − 0.781i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.763 - 0.645i$
Analytic conductor: \(8.19198\)
Root analytic conductor: \(8.19198\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1039, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1764,\ (0:\ ),\ 0.763 - 0.645i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.324146764 - 0.4846223859i\)
\(L(\frac12)\) \(\approx\) \(1.324146764 - 0.4846223859i\)
\(L(1)\) \(\approx\) \(1.007629706 - 0.1502205077i\)
\(L(1)\) \(\approx\) \(1.007629706 - 0.1502205077i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.623 - 0.781i)T \)
11 \( 1 + (0.222 + 0.974i)T \)
13 \( 1 + (0.733 - 0.680i)T \)
17 \( 1 + (-0.0747 - 0.997i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.900 - 0.433i)T \)
29 \( 1 + (0.826 - 0.563i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.826 - 0.563i)T \)
41 \( 1 + (-0.365 + 0.930i)T \)
43 \( 1 + (-0.365 - 0.930i)T \)
47 \( 1 + (-0.733 + 0.680i)T \)
53 \( 1 + (0.826 + 0.563i)T \)
59 \( 1 + (0.365 + 0.930i)T \)
61 \( 1 + (-0.0747 - 0.997i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (0.900 - 0.433i)T \)
73 \( 1 + (-0.955 + 0.294i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.733 - 0.680i)T \)
89 \( 1 + (0.733 + 0.680i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−20.15916377367236222830168354913, −19.39059619617037000381965868908, −18.98975215192763003496498051497, −18.26535887580310252907810844691, −17.3868986860862724634286768269, −16.55265079230046444817074731522, −15.90398641317493164422465270903, −15.03563655499592890369242770931, −14.570558171519159137661698641864, −13.56408194036574413601559211177, −13.04276052881876262632225738379, −11.86059314944209371533168141685, −11.208564371382581130067565913604, −10.828713494801419589896171024765, −9.811944302138947423893705002055, −8.68324130659169094699924786441, −8.3493148137640022629624698345, −7.174235935077217544998545163538, −6.55850215584538028770863453167, −5.85293864145914154058225889794, −4.638804589829837997592143110206, −3.76623604267461462208172234244, −3.167910117822786385179161010924, −2.09873601548001698666971226182, −0.869991545680840059816751434254, 0.69704312303510579093974934183, 1.65010932429435974847178455030, 2.853980579173338572346362817546, 3.8170381435288554688314423911, 4.62146029411434699039874072317, 5.2733211491857020548656611419, 6.34867680530688177390186587483, 7.23848266328979403469997265847, 8.01816169484091937114928837872, 8.71642993514136048661364392654, 9.49324055345538504887896743829, 10.36350171882966137134205567234, 11.22751044551854054644418310258, 12.0433872102734550208629660858, 12.64969961571065339898465796674, 13.27746498094998973874782225232, 14.289645136294926969259840429170, 15.11052027409372679056571072653, 15.72118117234293735755542579040, 16.464875957309739410579103012742, 17.11254652610206747703097387514, 18.0223275671822052694229952457, 18.61183484016363201957457159417, 19.64713463279258218916889495552, 20.13757282518010935184976038279

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