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Properties

Label 1-663-663.464-r0-0-0
Degree $1$
Conductor $663$
Sign $0.768 - 0.640i$
Analytic cond. $3.07895$
Root an. cond. $3.07895$
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  + (−0.965 − 0.258i)2-s + (0.866 + 0.5i)4-s + (0.923 + 0.382i)5-s + (0.130 − 0.991i)7-s + (−0.707 − 0.707i)8-s + (−0.793 − 0.608i)10-s + (−0.608 + 0.793i)11-s + (−0.382 + 0.923i)14-s + (0.5 + 0.866i)16-s + (−0.258 − 0.965i)19-s + (0.608 + 0.793i)20-s + (0.793 − 0.608i)22-s + (0.608 − 0.793i)23-s + (0.707 + 0.707i)25-s + (0.608 − 0.793i)28-s + (0.130 + 0.991i)29-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)2-s + (0.866 + 0.5i)4-s + (0.923 + 0.382i)5-s + (0.130 − 0.991i)7-s + (−0.707 − 0.707i)8-s + (−0.793 − 0.608i)10-s + (−0.608 + 0.793i)11-s + (−0.382 + 0.923i)14-s + (0.5 + 0.866i)16-s + (−0.258 − 0.965i)19-s + (0.608 + 0.793i)20-s + (0.793 − 0.608i)22-s + (0.608 − 0.793i)23-s + (0.707 + 0.707i)25-s + (0.608 − 0.793i)28-s + (0.130 + 0.991i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 663 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.768 - 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 663 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.768 - 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(663\)    =    \(3 \cdot 13 \cdot 17\)
Sign: $0.768 - 0.640i$
Analytic conductor: \(3.07895\)
Root analytic conductor: \(3.07895\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{663} (464, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 663,\ (0:\ ),\ 0.768 - 0.640i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9983189906 - 0.3613343090i\)
\(L(\frac12)\) \(\approx\) \(0.9983189906 - 0.3613343090i\)
\(L(1)\) \(\approx\) \(0.8350715869 - 0.1396519250i\)
\(L(1)\) \(\approx\) \(0.8350715869 - 0.1396519250i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.965 - 0.258i)T \)
5 \( 1 + (0.923 + 0.382i)T \)
7 \( 1 + (0.130 - 0.991i)T \)
11 \( 1 + (-0.608 + 0.793i)T \)
19 \( 1 + (-0.258 - 0.965i)T \)
23 \( 1 + (0.608 - 0.793i)T \)
29 \( 1 + (0.130 + 0.991i)T \)
31 \( 1 + (0.382 - 0.923i)T \)
37 \( 1 + (0.991 - 0.130i)T \)
41 \( 1 + (-0.793 - 0.608i)T \)
43 \( 1 + (0.965 - 0.258i)T \)
47 \( 1 - iT \)
53 \( 1 + (0.707 - 0.707i)T \)
59 \( 1 + (0.965 - 0.258i)T \)
61 \( 1 + (-0.130 + 0.991i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (0.608 + 0.793i)T \)
73 \( 1 + (-0.923 - 0.382i)T \)
79 \( 1 + (0.382 + 0.923i)T \)
83 \( 1 + (-0.707 + 0.707i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (-0.793 + 0.608i)T \)
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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

−23.05405794910717118147646063328, −21.65924733128842618039778181624, −21.25631699626809998543215712347, −20.50497596420825093197392407044, −19.32196751687709475469144168596, −18.69495189851700373043044978720, −17.975836228719653158307018942736, −17.2309574766263436856756554537, −16.39383179125599898354966582789, −15.67232319086080730541108002389, −14.77337433937647635041948928867, −13.84436736711044637658037435423, −12.80219842470979075880052446787, −11.84836502647346526029065438820, −10.91802602859297694892426294056, −10.00194939451171408206354458857, −9.25007933592636625341929828139, −8.4851415453129335978122086571, −7.76509616416422873080108645683, −6.31269909653289976964651994365, −5.82312956633888458649440977, −4.96583604557609965171000004524, −3.03520470809621519164796195203, −2.17183383594264247671043357401, −1.12606497898399434199693080950, 0.833291072254444469163806603422, 2.07974130386822974764350709215, 2.82529150774394586888320965977, 4.18759046465157251081528086611, 5.428558982130161937962762002668, 6.82393571919899661675172540253, 7.087307274185451999672270751360, 8.2824028855343598203613622439, 9.27268177466408878033655425029, 10.14061154061412579335714861663, 10.60596937138364453806271887741, 11.45472811139214286747945935512, 12.78157331020399033652931375236, 13.32936157233464835990511292762, 14.52257054038966894236759426155, 15.31642815847031706787649232919, 16.461281184301830035681007769638, 17.156409058544393971852644442447, 17.7984887934284283917007324630, 18.436342418573864187446690687712, 19.40582729068556245765232441589, 20.36124708329998303755699600056, 20.796338595109106978365547148685, 21.69860883449543625883061256538, 22.58424845815287877809955366281

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