Properties

Label 1-42e2-1764.1343-r1-0-0
Degree $1$
Conductor $1764$
Sign $-0.915 - 0.401i$
Analytic cond. $189.568$
Root an. cond. $189.568$
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  + (0.0747 − 0.997i)5-s + (0.365 − 0.930i)11-s + (0.988 − 0.149i)13-s + (−0.222 − 0.974i)17-s + 19-s + (−0.733 − 0.680i)23-s + (−0.988 − 0.149i)25-s + (0.733 − 0.680i)29-s + (−0.5 + 0.866i)31-s + (−0.222 − 0.974i)37-s + (0.0747 − 0.997i)41-s + (−0.0747 − 0.997i)43-s + (−0.365 + 0.930i)47-s + (0.222 − 0.974i)53-s + (−0.900 − 0.433i)55-s + ⋯
L(s)  = 1  + (0.0747 − 0.997i)5-s + (0.365 − 0.930i)11-s + (0.988 − 0.149i)13-s + (−0.222 − 0.974i)17-s + 19-s + (−0.733 − 0.680i)23-s + (−0.988 − 0.149i)25-s + (0.733 − 0.680i)29-s + (−0.5 + 0.866i)31-s + (−0.222 − 0.974i)37-s + (0.0747 − 0.997i)41-s + (−0.0747 − 0.997i)43-s + (−0.365 + 0.930i)47-s + (0.222 − 0.974i)53-s + (−0.900 − 0.433i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.915 - 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.915 - 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.915 - 0.401i$
Analytic conductor: \(189.568\)
Root analytic conductor: \(189.568\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1764,\ (1:\ ),\ -0.915 - 0.401i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4219555901 - 2.013338078i\)
\(L(\frac12)\) \(\approx\) \(0.4219555901 - 2.013338078i\)
\(L(1)\) \(\approx\) \(1.037695287 - 0.4898114736i\)
\(L(1)\) \(\approx\) \(1.037695287 - 0.4898114736i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.0747 - 0.997i)T \)
11 \( 1 + (0.365 - 0.930i)T \)
13 \( 1 + (0.988 - 0.149i)T \)
17 \( 1 + (-0.222 - 0.974i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.733 - 0.680i)T \)
29 \( 1 + (0.733 - 0.680i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.222 - 0.974i)T \)
41 \( 1 + (0.0747 - 0.997i)T \)
43 \( 1 + (-0.0747 - 0.997i)T \)
47 \( 1 + (-0.365 + 0.930i)T \)
53 \( 1 + (0.222 - 0.974i)T \)
59 \( 1 + (-0.826 + 0.563i)T \)
61 \( 1 + (0.733 - 0.680i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.222 + 0.974i)T \)
73 \( 1 + (-0.623 + 0.781i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.988 + 0.149i)T \)
89 \( 1 + (0.623 - 0.781i)T \)
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

−20.23618084057083177202628793418, −19.70994507072984307541690175848, −18.816087075421266889669387216550, −18.05507766163293898260812983755, −17.72723279044693665439026822043, −16.696357884161684290506301974980, −15.83067818689563391387600623579, −15.12570083795887604603187852455, −14.53864651143539761750239866369, −13.700973426815172735020230730860, −13.071349477347904691928493508122, −11.95296848481185481655471891862, −11.43819444396297237387160815301, −10.5291357967938747941382377034, −9.91975292838766483039853584184, −9.118060050975985369194979758725, −8.057656519471076753792411761478, −7.37821069527137733389627662142, −6.45905349823645720437460744332, −5.98629436299125548720351656648, −4.789611039017661961706389396191, −3.813570067675498538789725836563, −3.1848911674944352537729504459, −2.02847477656914131704175407626, −1.28350188646637346502105809162, 0.41360313561777075361253343322, 1.02297161313958828497820969557, 2.133166069141681076378318195706, 3.32344231069152761100102984176, 4.05404275600919569630150306977, 5.07693621639851975780667326894, 5.72159874023963181537319081613, 6.55720403965244250677682720935, 7.61887223061647091807735184036, 8.5045582041990089047641727770, 8.95392445620093357240114977973, 9.79251757318425569268467718728, 10.77158030533189743828871195915, 11.58103009488214439519508631394, 12.20914785503713655593333135724, 13.06403343299843694571384216683, 13.92243864046977614090337650919, 14.15353414990429452615227943867, 15.797261522402714686109734385130, 15.907747025046557265718533733108, 16.64174491558774184928012699975, 17.58403244956204510057325581599, 18.18969716765114787499693165813, 19.03046937692857193989523597270, 19.87757526967983062689587183330

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