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Properties

Label 1-42e2-1764.1679-r1-0-0
Degree $1$
Conductor $1764$
Sign $0.0676 - 0.997i$
Analytic cond. $189.568$
Root an. cond. $189.568$
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  + (0.955 − 0.294i)5-s + (0.0747 − 0.997i)11-s + (−0.826 − 0.563i)13-s + (0.623 + 0.781i)17-s + 19-s + (−0.988 − 0.149i)23-s + (0.826 − 0.563i)25-s + (0.988 − 0.149i)29-s + (−0.5 − 0.866i)31-s + (0.623 + 0.781i)37-s + (0.955 − 0.294i)41-s + (−0.955 − 0.294i)43-s + (−0.0747 + 0.997i)47-s + (−0.623 + 0.781i)53-s + (−0.222 − 0.974i)55-s + ⋯
L(s)  = 1  + (0.955 − 0.294i)5-s + (0.0747 − 0.997i)11-s + (−0.826 − 0.563i)13-s + (0.623 + 0.781i)17-s + 19-s + (−0.988 − 0.149i)23-s + (0.826 − 0.563i)25-s + (0.988 − 0.149i)29-s + (−0.5 − 0.866i)31-s + (0.623 + 0.781i)37-s + (0.955 − 0.294i)41-s + (−0.955 − 0.294i)43-s + (−0.0747 + 0.997i)47-s + (−0.623 + 0.781i)53-s + (−0.222 − 0.974i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0676 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0676 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.0676 - 0.997i$
Analytic conductor: \(189.568\)
Root analytic conductor: \(189.568\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1679, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1764,\ (1:\ ),\ 0.0676 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.890830601 - 1.767009293i\)
\(L(\frac12)\) \(\approx\) \(1.890830601 - 1.767009293i\)
\(L(1)\) \(\approx\) \(1.265283972 - 0.2802253157i\)
\(L(1)\) \(\approx\) \(1.265283972 - 0.2802253157i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.955 - 0.294i)T \)
11 \( 1 + (0.0747 - 0.997i)T \)
13 \( 1 + (-0.826 - 0.563i)T \)
17 \( 1 + (0.623 + 0.781i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.988 - 0.149i)T \)
29 \( 1 + (0.988 - 0.149i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.623 + 0.781i)T \)
41 \( 1 + (0.955 - 0.294i)T \)
43 \( 1 + (-0.955 - 0.294i)T \)
47 \( 1 + (-0.0747 + 0.997i)T \)
53 \( 1 + (-0.623 + 0.781i)T \)
59 \( 1 + (0.733 - 0.680i)T \)
61 \( 1 + (0.988 - 0.149i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (0.623 - 0.781i)T \)
73 \( 1 + (0.900 + 0.433i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (-0.826 + 0.563i)T \)
89 \( 1 + (-0.900 - 0.433i)T \)
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

−20.1094805029135578917441480269, −19.70339462948984440165099664974, −18.51182505434337695647062570523, −18.00757234989961960452832790592, −17.47347647864387544304050517699, −16.51542800843731929755809920289, −15.957908081257105390299371217305, −14.79044647933433562033570552720, −14.30107728648405778854507868384, −13.72655104255845599723398577436, −12.702190799173851660332807952127, −12.07247314365330882117510070808, −11.287739341284661384617587156587, −10.08610917979689912774071676503, −9.81201911637463195187118010813, −9.12507710557370775469636670601, −7.90833363259890058141420913806, −7.10529705832258474942148014002, −6.53690403314238910448351535192, −5.3758555097812529152127586992, −4.934616080907691422939071206023, −3.764763390254502151939150084608, −2.67429628754117814536183672008, −2.019440323396859797382909324021, −1.011295699632290668159696317987, 0.49699480645340366218721562679, 1.36966648830939935305891341732, 2.44911320861255005925371155713, 3.22637098869433952403986099246, 4.32072638761074363778408168251, 5.383442290579337144559379542, 5.83167921520527021557843190305, 6.663548223531290334543169988391, 7.868223159540498741932374272196, 8.349726487419146653339642165131, 9.5010257085495553902888387434, 9.913580684858882745187377190217, 10.741667031454116888175835070964, 11.69080460772775822714046936704, 12.50863595372727068893452117729, 13.162946864593132570708911613, 14.06059044585984670770638076839, 14.41487153702840158931200930662, 15.50743485726643599453523138801, 16.3348912494379745379452081401, 16.957281626110027374204333072615, 17.62535239114288309803586860029, 18.34635577022708009352408147016, 19.127196577627962117458357082134, 19.9668854449087545582749906031

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