Properties

Label 1-42e2-1764.1399-r0-0-0
Degree $1$
Conductor $1764$
Sign $-0.401 - 0.915i$
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  + (−0.0747 − 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

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

Invariants

Degree: \(1\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.401 - 0.915i$
Analytic conductor: \(8.19198\)
Root analytic conductor: \(8.19198\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1399, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1764,\ (0:\ ),\ -0.401 - 0.915i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8016523672 - 1.226769145i\)
\(L(\frac12)\) \(\approx\) \(0.8016523672 - 1.226769145i\)
\(L(1)\) \(\approx\) \(1.003490294 - 0.3976464919i\)
\(L(1)\) \(\approx\) \(1.003490294 - 0.3976464919i\)

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

−20.39588734288242883410837227197, −19.773084833861472136835419287727, −18.868637962057068597946014119693, −18.247348518447726400216281232251, −17.74310542195422407271190690598, −16.854312949380459619361475166961, −15.82451951983604071263834951481, −15.32082838822546215715336101976, −14.60007216588847777648441904816, −13.86309559074094384618255930132, −13.020992718258727662724658651, −12.30194640661844750701842056908, −11.28031726263253811752830468685, −10.762771605776456346586152693201, −10.02119332175738078991494458844, −9.19750170833825070582232375312, −8.20459263404076243283427399690, −7.34802251479551162615898923956, −6.84188294245456323040240570298, −5.79750493644772657473732449506, −5.12137165057294293602718331291, −3.76995628532709102733567509327, −3.37087973936514393905953553768, −2.21699033468736502966823498040, −1.33283300807057579111288830174, 0.570287176914690905241796816719, 1.35856554345750357226954645207, 2.65451464054544040425349342869, 3.55716826635802058984779245889, 4.44820775808596770747151729707, 5.39516479182214564532980139124, 5.87922567200364586551278229581, 7.04984676297429682263967696782, 7.94102941288072845502936209576, 8.62958326938741229440843187494, 9.28978581443425745231644023158, 10.093033458135295038157116183663, 11.334611391744548236597314383023, 11.51895263234722446796319217431, 12.66826123372357772983866651843, 13.36700238018348195649231788936, 13.81205009975674138222847910748, 14.84823464369174988332502129230, 15.95663127040711328823978753461, 16.14972473204838561165583656662, 16.9219197424572653940113192636, 17.804986115186505976847525740257, 18.69440680845197899073860959894, 19.09131053430316602697285117736, 20.29913412504959780638504014052

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