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Properties

Label 1-42e2-1764.535-r0-0-0
Degree $1$
Conductor $1764$
Sign $-0.592 + 0.805i$
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  + (0.900 + 0.433i)5-s + (−0.623 + 0.781i)11-s + (0.988 + 0.149i)13-s + (−0.955 + 0.294i)17-s + (−0.5 + 0.866i)19-s + (0.222 + 0.974i)23-s + (0.623 + 0.781i)25-s + (−0.733 − 0.680i)29-s + (−0.5 + 0.866i)31-s + (−0.733 − 0.680i)37-s + (−0.0747 − 0.997i)41-s + (−0.0747 + 0.997i)43-s + (−0.988 − 0.149i)47-s + (−0.733 + 0.680i)53-s + (−0.900 + 0.433i)55-s + ⋯
L(s)  = 1  + (0.900 + 0.433i)5-s + (−0.623 + 0.781i)11-s + (0.988 + 0.149i)13-s + (−0.955 + 0.294i)17-s + (−0.5 + 0.866i)19-s + (0.222 + 0.974i)23-s + (0.623 + 0.781i)25-s + (−0.733 − 0.680i)29-s + (−0.5 + 0.866i)31-s + (−0.733 − 0.680i)37-s + (−0.0747 − 0.997i)41-s + (−0.0747 + 0.997i)43-s + (−0.988 − 0.149i)47-s + (−0.733 + 0.680i)53-s + (−0.900 + 0.433i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.592 + 0.805i$
Analytic conductor: \(8.19198\)
Root analytic conductor: \(8.19198\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (535, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1764,\ (0:\ ),\ -0.592 + 0.805i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5664370290 + 1.119570546i\)
\(L(\frac12)\) \(\approx\) \(0.5664370290 + 1.119570546i\)
\(L(1)\) \(\approx\) \(1.023305744 + 0.3231831351i\)
\(L(1)\) \(\approx\) \(1.023305744 + 0.3231831351i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.900 + 0.433i)T \)
11 \( 1 + (-0.623 + 0.781i)T \)
13 \( 1 + (0.988 + 0.149i)T \)
17 \( 1 + (-0.955 + 0.294i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.222 + 0.974i)T \)
29 \( 1 + (-0.733 - 0.680i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.733 - 0.680i)T \)
41 \( 1 + (-0.0747 - 0.997i)T \)
43 \( 1 + (-0.0747 + 0.997i)T \)
47 \( 1 + (-0.988 - 0.149i)T \)
53 \( 1 + (-0.733 + 0.680i)T \)
59 \( 1 + (0.0747 - 0.997i)T \)
61 \( 1 + (-0.955 + 0.294i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (0.222 + 0.974i)T \)
73 \( 1 + (-0.365 + 0.930i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.988 + 0.149i)T \)
89 \( 1 + (0.988 - 0.149i)T \)
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

−20.25706164567744909350813558555, −19.152493500354963338281061917536, −18.35065688300799592872745274626, −17.91709659417044426175123473775, −16.96167731775806136948417284576, −16.38310733263645303264532241726, −15.63218475667991080556984463223, −14.79502059068317673880149086447, −13.82744984268123325925599041948, −13.17614951967331028536279985861, −12.92503511924640585452038852472, −11.608556329595029308284326910014, −10.86088992340366372549880429083, −10.321918127683139445555912787919, −9.112268106559181004617527488162, −8.80531576244100177127569178832, −7.92612166706878765562412494603, −6.68206312331403438469662000780, −6.16918096342685355660333931835, −5.23759439669438302870254256554, −4.57745102779330097387915325451, −3.38566977415677520761952737394, −2.492210847635419922835034574026, −1.60140149471107225015447165953, −0.404845640349582586125616547, 1.594744955954682800244526832362, 2.042128871370215493190655976357, 3.19831535748430662285039931439, 4.06457151500287561344456277608, 5.14420636515866608284168927164, 5.888593465661733870560579837454, 6.6287457058443431072333060261, 7.42652121302791577580381559852, 8.397951341232968601598631217317, 9.24411009383502616186317676359, 9.94575611721837074414906403312, 10.76443982601131854430284119260, 11.24556183162877533421444003711, 12.52300489925059088964410299068, 13.06417233153253811833901640897, 13.79354868011789260376047195186, 14.51354246971590950744773281959, 15.3665075667387313277087412765, 15.93541362269720473601934880600, 17.0324868752511287233336482820, 17.59383432477880412025424936421, 18.25498554041200261009497261788, 18.85800056965067100860060236410, 19.79560569289444454218387881120, 20.67577437278746504435886045701

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