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Properties

Label 1-663-663.641-r0-0-0
Degree $1$
Conductor $663$
Sign $0.713 + 0.701i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 663 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.713 + 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 663 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.713 + 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(663\)    =    \(3 \cdot 13 \cdot 17\)
Sign: $0.713 + 0.701i$
Analytic conductor: \(3.07895\)
Root analytic conductor: \(3.07895\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{663} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 663,\ (0:\ ),\ 0.713 + 0.701i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.875974863 + 1.177008853i\)
\(L(\frac12)\) \(\approx\) \(2.875974863 + 1.177008853i\)
\(L(1)\) \(\approx\) \(2.092616484 + 0.5174074409i\)
\(L(1)\) \(\approx\) \(2.092616484 + 0.5174074409i\)

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

−22.42102693371693929088947339349, −21.71287541010016459108725840734, −21.41056065209504373475381945883, −20.47058147881914980813371272008, −19.676152022182044347421522785741, −18.54998177630147462227769903972, −17.98527557693510212580498010154, −16.651209298231739851757008926700, −16.00985321971525298117818539822, −15.11930769241523775408692207348, −14.22574432673376445116773124081, −13.491015378350803026022337742148, −12.76196597965418991984989679431, −11.99299148652801350955234040388, −11.01890279789478349859427748543, −10.2006065348609585750699860755, −9.1461560846922623442686684480, −8.28371324004321420684567065323, −6.85445790163617122624484335945, −5.93616522737289262204476469306, −5.33766035368438116803924732544, −4.527735518078145976232044233075, −2.98811219074367600571332462852, −2.43349682436346719999473079724, −1.23456190454553645161737526231, 1.60674114492321205692780118958, 2.49311456907923372746733505589, 3.675056111795353728890358249879, 4.5529581269987994758139328369, 5.60183882364511753538334961402, 6.325075740162429580628478277903, 7.401283260390136116795838942313, 7.899225440425768514557452945609, 9.639840071291702299077028481564, 10.29741149565784571899540927147, 11.16026527928183714115179472103, 12.20490614024336462957268775523, 13.19503793747779919224921403184, 13.70658835054696852022255067458, 14.46124441868159259170085074215, 15.26111510967973320741555920500, 16.23503778078473116539494945905, 17.16435200226741117859819711397, 17.65845918954300118727412948970, 18.784986595893237144819604958077, 20.01756049102764492579590314513, 20.72023079181675745950025640201, 21.215322138780017083499918535490, 22.27434889456549294407715672213, 22.86413054103973088073937214025

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