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Properties

Label 1-42e2-1764.1231-r0-0-0
Degree $1$
Conductor $1764$
Sign $0.999 + 0.00356i$
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  + (0.988 − 0.149i)5-s + (0.733 − 0.680i)11-s + (−0.955 − 0.294i)13-s + (0.900 + 0.433i)17-s + 19-s + (−0.0747 + 0.997i)23-s + (0.955 − 0.294i)25-s + (0.0747 + 0.997i)29-s + (−0.5 + 0.866i)31-s + (−0.900 − 0.433i)37-s + (0.988 − 0.149i)41-s + (0.988 + 0.149i)43-s + (−0.733 + 0.680i)47-s + (−0.900 + 0.433i)53-s + (0.623 − 0.781i)55-s + ⋯
L(s)  = 1  + (0.988 − 0.149i)5-s + (0.733 − 0.680i)11-s + (−0.955 − 0.294i)13-s + (0.900 + 0.433i)17-s + 19-s + (−0.0747 + 0.997i)23-s + (0.955 − 0.294i)25-s + (0.0747 + 0.997i)29-s + (−0.5 + 0.866i)31-s + (−0.900 − 0.433i)37-s + (0.988 − 0.149i)41-s + (0.988 + 0.149i)43-s + (−0.733 + 0.680i)47-s + (−0.900 + 0.433i)53-s + (0.623 − 0.781i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00356i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00356i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.999 + 0.00356i$
Analytic conductor: \(8.19198\)
Root analytic conductor: \(8.19198\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1231, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1764,\ (0:\ ),\ 0.999 + 0.00356i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.125054184 + 0.003784615460i\)
\(L(\frac12)\) \(\approx\) \(2.125054184 + 0.003784615460i\)
\(L(1)\) \(\approx\) \(1.361267794 + 0.02650407272i\)
\(L(1)\) \(\approx\) \(1.361267794 + 0.02650407272i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.988 - 0.149i)T \)
11 \( 1 + (0.733 - 0.680i)T \)
13 \( 1 + (-0.955 - 0.294i)T \)
17 \( 1 + (0.900 + 0.433i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.0747 + 0.997i)T \)
29 \( 1 + (0.0747 + 0.997i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.900 - 0.433i)T \)
41 \( 1 + (0.988 - 0.149i)T \)
43 \( 1 + (0.988 + 0.149i)T \)
47 \( 1 + (-0.733 + 0.680i)T \)
53 \( 1 + (-0.900 + 0.433i)T \)
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 \( 1 + (0.222 - 0.974i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.955 - 0.294i)T \)
89 \( 1 + (0.222 - 0.974i)T \)
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

−20.47145962008818843122486940217, −19.43673631734698911094408217481, −18.75136204820137921622933737600, −17.95339088826363524446031125291, −17.28090114407618357107614813653, −16.74372767044643229457650232602, −15.89844542164558747413236333131, −14.729585649656531599750512042846, −14.412623912075700772301311085374, −13.68052823403470342671652594916, −12.72011740664869598432079247771, −12.07906075723743805225638433938, −11.311635038827119992566504711577, −10.16454694942514260739384629978, −9.69350960310777842114066338751, −9.16895405175842626534114112133, −7.95328178509070366521196196419, −7.13658588164009453804365180105, −6.45475433313995900295374417625, −5.51400706771980906131919846980, −4.830854219553785374670308971064, −3.82602613948613225098956705593, −2.678590464535469146558049653, −2.02336160531850666609362686426, −0.93836454916092545715568252442, 1.03305231425999118227662317851, 1.770471232133332599844650265377, 2.96616267714645189857961147318, 3.61522085321244590351596240854, 4.98077262405931841054606118780, 5.50070101441456106114000464866, 6.289648170702314364480246839834, 7.23295122576485808067120694733, 8.00848692852892677975496322082, 9.20077362507769824110915343218, 9.4462561491000060149309929810, 10.42798829003882831144472982220, 11.12546747173521233564205922064, 12.26661546952366311886532226533, 12.60233068210080852780045401472, 13.80355493602766185144048315512, 14.15890055761286337283468902600, 14.84963097817918513830454979114, 16.02884151659079126670013372988, 16.55627179518053887439830640879, 17.466564403566994145498201283230, 17.7711051414658883212538092194, 18.81535295681837111514154350714, 19.54625509047507594573447778277, 20.171455766085127339740336649688

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