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Properties

Label 1-42e2-1764.11-r0-0-0
Degree $1$
Conductor $1764$
Sign $-0.960 + 0.277i$
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  + (−0.955 + 0.294i)5-s + (0.0747 − 0.997i)11-s + (0.0747 − 0.997i)13-s + (−0.365 + 0.930i)17-s + (0.5 − 0.866i)19-s + (−0.988 − 0.149i)23-s + (0.826 − 0.563i)25-s + (−0.365 + 0.930i)29-s − 31-s + (−0.988 + 0.149i)37-s + (0.733 + 0.680i)41-s + (0.733 − 0.680i)43-s + (−0.900 + 0.433i)47-s + (0.988 + 0.149i)53-s + (0.222 + 0.974i)55-s + ⋯
L(s)  = 1  + (−0.955 + 0.294i)5-s + (0.0747 − 0.997i)11-s + (0.0747 − 0.997i)13-s + (−0.365 + 0.930i)17-s + (0.5 − 0.866i)19-s + (−0.988 − 0.149i)23-s + (0.826 − 0.563i)25-s + (−0.365 + 0.930i)29-s − 31-s + (−0.988 + 0.149i)37-s + (0.733 + 0.680i)41-s + (0.733 − 0.680i)43-s + (−0.900 + 0.433i)47-s + (0.988 + 0.149i)53-s + (0.222 + 0.974i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.960 + 0.277i$
Analytic conductor: \(8.19198\)
Root analytic conductor: \(8.19198\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1764,\ (0:\ ),\ -0.960 + 0.277i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001498855127 + 0.01058284259i\)
\(L(\frac12)\) \(\approx\) \(0.001498855127 + 0.01058284259i\)
\(L(1)\) \(\approx\) \(0.7238266255 - 0.03658385201i\)
\(L(1)\) \(\approx\) \(0.7238266255 - 0.03658385201i\)

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 \( 1 + (0.0747 - 0.997i)T \)
17 \( 1 + (-0.365 + 0.930i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.988 - 0.149i)T \)
29 \( 1 + (-0.365 + 0.930i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.988 + 0.149i)T \)
41 \( 1 + (0.733 + 0.680i)T \)
43 \( 1 + (0.733 - 0.680i)T \)
47 \( 1 + (-0.900 + 0.433i)T \)
53 \( 1 + (0.988 + 0.149i)T \)
59 \( 1 + (-0.222 - 0.974i)T \)
61 \( 1 + (0.623 + 0.781i)T \)
67 \( 1 - T \)
71 \( 1 + (0.623 - 0.781i)T \)
73 \( 1 + (0.0747 + 0.997i)T \)
79 \( 1 - T \)
83 \( 1 + (0.0747 + 0.997i)T \)
89 \( 1 + (-0.826 + 0.563i)T \)
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

−19.91605836708053941255732414221, −19.25655522907888276684387508528, −18.44327080878244664384620773371, −17.809839009595698678426164806462, −16.8001957781989687202463805580, −16.14281118809375426528644574951, −15.62694703642471774756090128683, −14.69813036476492117034422100141, −14.07893055954579621549810929148, −13.13214641670986267056944713863, −12.196353834741890790123805944309, −11.82148460527329977927941030638, −11.05981299710946107201091780832, −9.96578211648945794028136042486, −9.3081656151133347809972840647, −8.501985704138387278328048873578, −7.466361798992200078364725845022, −7.18564134789146162451565447006, −6.01459855166666008965797900491, −5.009431888995527384510058810157, −4.20517535513190344789752848301, −3.66040780327667410580407740120, −2.35625080000399716157304018881, −1.48314741963188064319879851149, −0.00410438704413486557908908289, 1.17877867201388416071477039221, 2.557056326606517916866366785730, 3.424276644844034557267990589751, 3.98838961683355384177323574423, 5.13809965863961210906181828604, 5.93116406405755721578577058292, 6.85883261758239695494379810393, 7.67910447489969135929074421900, 8.378641654951001419811355013550, 9.017599390178394160512873727079, 10.24100521200239611160838229174, 10.919968191600791005411477540301, 11.42428698129937536237396286025, 12.4156027693495185158692391651, 13.01308653659723724599551614086, 13.97601943721509863555047172278, 14.716811921198885021881384233, 15.48773926103771533524149742746, 16.03234269533771775180564640882, 16.77279785777344158821454167219, 17.81634462547351034897675485676, 18.31648777969706080900638685278, 19.27304920384705383861982177047, 19.76942654648664623721153677757, 20.355713361792099149997533485259

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