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Properties

Label 1-42e2-1764.1271-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  + (0.988 + 0.149i)5-s + (−0.733 − 0.680i)11-s + (−0.733 − 0.680i)13-s + (−0.826 − 0.563i)17-s + (0.5 − 0.866i)19-s + (0.0747 + 0.997i)23-s + (0.955 + 0.294i)25-s + (−0.826 − 0.563i)29-s − 31-s + (0.0747 − 0.997i)37-s + (−0.365 − 0.930i)41-s + (−0.365 + 0.930i)43-s + (−0.222 + 0.974i)47-s + (−0.0747 − 0.997i)53-s + (−0.623 − 0.781i)55-s + ⋯
L(s)  = 1  + (0.988 + 0.149i)5-s + (−0.733 − 0.680i)11-s + (−0.733 − 0.680i)13-s + (−0.826 − 0.563i)17-s + (0.5 − 0.866i)19-s + (0.0747 + 0.997i)23-s + (0.955 + 0.294i)25-s + (−0.826 − 0.563i)29-s − 31-s + (0.0747 − 0.997i)37-s + (−0.365 − 0.930i)41-s + (−0.365 + 0.930i)43-s + (−0.222 + 0.974i)47-s + (−0.0747 − 0.997i)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: $\chi_{1764} (1271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1764,\ (0:\ ),\ -0.763 - 0.645i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2688573309 - 0.7346061084i\)
\(L(\frac12)\) \(\approx\) \(0.2688573309 - 0.7346061084i\)
\(L(1)\) \(\approx\) \(0.9408195790 - 0.1751238847i\)
\(L(1)\) \(\approx\) \(0.9408195790 - 0.1751238847i\)

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 \( 1 + (-0.733 - 0.680i)T \)
17 \( 1 + (-0.826 - 0.563i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.0747 + 0.997i)T \)
29 \( 1 + (-0.826 - 0.563i)T \)
31 \( 1 - T \)
37 \( 1 + (0.0747 - 0.997i)T \)
41 \( 1 + (-0.365 - 0.930i)T \)
43 \( 1 + (-0.365 + 0.930i)T \)
47 \( 1 + (-0.222 + 0.974i)T \)
53 \( 1 + (-0.0747 - 0.997i)T \)
59 \( 1 + (0.623 + 0.781i)T \)
61 \( 1 + (-0.900 + 0.433i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.900 - 0.433i)T \)
73 \( 1 + (-0.733 + 0.680i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.733 + 0.680i)T \)
89 \( 1 + (-0.955 - 0.294i)T \)
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

−20.34487783052926554673274678956, −20.16400800603804905584400044445, −18.776102306844420991281353029984, −18.38816562004873727877830535384, −17.58015484028187586563108617432, −16.81455244264997082365172697118, −16.348905696707713435663987551, −15.15227592601302126558463459303, −14.64945958495985452473311447775, −13.81440921545191468918021816546, −13.02083173032056641013700092846, −12.50719812809845543283197549097, −11.57510972566892373830643211054, −10.49560100336750669213822107338, −10.06026034242620592528946927671, −9.2190755124642008610203627983, −8.50576219624161678821034515110, −7.436863893948278131305117386785, −6.728568835657699076649696184788, −5.85236213033123255807088064976, −5.02869912693522734540107202509, −4.359510458946553393340596598311, −3.10390133018175952726767293058, −2.10577857277618541699834761587, −1.59375737919534938127184640013, 0.24405903581539278533640230963, 1.62579432214448390757471815084, 2.600593757510471438560272108087, 3.16332861024029406721996087609, 4.51237005962864021957048624635, 5.47552968353822898726424469311, 5.77662584190060788263808250814, 7.055690769985138652472937222487, 7.539952237109711299630866485188, 8.72513474281252814380050183640, 9.39498494628279627109635562828, 10.07946767459662206598828598212, 10.96690158991727835243873325801, 11.500310793177636906007943379801, 12.79042466699170049696686096352, 13.24818304480897481005215335838, 13.87926985182952798250883047627, 14.75157648981605451417524032910, 15.52614900967545951334402595362, 16.26007135522766034555634462102, 17.1434927062204988888161817390, 17.88216379604526442139650402001, 18.19911787768288934237653413471, 19.28409127768898322051411187673, 19.93276431849045902945008080909

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