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Properties

Label 1-42e2-1764.743-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: $\chi_{1764} (743, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1764,\ (0:\ ),\ 0.999 + 0.00356i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.022408853 + 0.003601809248i\)
\(L(\frac12)\) \(\approx\) \(2.022408853 + 0.003601809248i\)
\(L(1)\) \(\approx\) \(1.316048379 + 0.001574151420i\)
\(L(1)\) \(\approx\) \(1.316048379 + 0.001574151420i\)

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

−20.567474313770556221232984358827, −19.35403909058587657762799147403, −18.74164992865981701324905513893, −17.97880148865126221365924831399, −17.34687014111935592156338580013, −16.63202405390121800329374985031, −15.88093336317424398054666872857, −14.95308501989971722543496967265, −14.322533718970053319128767763079, −13.43796947872058990718058106690, −12.879905235182366410011598568939, −12.207066642878238865392320866962, −11.04817199655077056276307627767, −10.36672270271383391152270371840, −9.78585717070427018780289856145, −8.8430608476649078045150084961, −8.17240277268745509882017452150, −7.19250906968330290708900310808, −6.167150851431962369307623665935, −5.7715655163388399976678641040, −4.677531539731219506837396694934, −3.93799047766356714240487391403, −2.618922833087496657974232749312, −2.03358146837238968437360149077, −0.92921836035687738662597472593, 0.949348713843591961359720381051, 1.884640310736240817976045811756, 2.955225983766376768316955427508, 3.56652039565179247315161485638, 4.96123249591108849797757367753, 5.58514743641143115206031199509, 6.265800502079766701382758259552, 7.163077737446408815873863889391, 8.18034349366152813932368467378, 8.84411393864080414168055610365, 9.7034999483645569921224712199, 10.59360566249136906396014139221, 10.92306734731245316717673269597, 12.096885312947810216679972271976, 12.958948715471093912026428607330, 13.56060107789680826808029796323, 14.16110948739000884734764581314, 15.00523924971674259146768083753, 15.96387545605496456417359603163, 16.46684053454133872767357218781, 17.47445717011094935765746763655, 17.95855109489374413943811481839, 18.763967011568258839641289349387, 19.30987880092455287032866414075, 20.48962234428946760088584314751

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