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Properties

Label 1-555-555.254-r0-0-0
Degree $1$
Conductor $555$
Sign $0.437 + 0.899i$
Analytic cond. $2.57740$
Root an. cond. $2.57740$
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  + (0.642 + 0.766i)2-s + (−0.173 + 0.984i)4-s + (0.939 − 0.342i)7-s + (−0.866 + 0.5i)8-s + (−0.5 − 0.866i)11-s + (0.984 + 0.173i)13-s + (0.866 + 0.5i)14-s + (−0.939 − 0.342i)16-s + (0.984 − 0.173i)17-s + (0.642 − 0.766i)19-s + (0.342 − 0.939i)22-s + (0.866 + 0.5i)23-s + (0.5 + 0.866i)26-s + (0.173 + 0.984i)28-s + (−0.866 + 0.5i)29-s + ⋯
L(s)  = 1  + (0.642 + 0.766i)2-s + (−0.173 + 0.984i)4-s + (0.939 − 0.342i)7-s + (−0.866 + 0.5i)8-s + (−0.5 − 0.866i)11-s + (0.984 + 0.173i)13-s + (0.866 + 0.5i)14-s + (−0.939 − 0.342i)16-s + (0.984 − 0.173i)17-s + (0.642 − 0.766i)19-s + (0.342 − 0.939i)22-s + (0.866 + 0.5i)23-s + (0.5 + 0.866i)26-s + (0.173 + 0.984i)28-s + (−0.866 + 0.5i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(555\)    =    \(3 \cdot 5 \cdot 37\)
Sign: $0.437 + 0.899i$
Analytic conductor: \(2.57740\)
Root analytic conductor: \(2.57740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{555} (254, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 555,\ (0:\ ),\ 0.437 + 0.899i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.829085765 + 1.144028519i\)
\(L(\frac12)\) \(\approx\) \(1.829085765 + 1.144028519i\)
\(L(1)\) \(\approx\) \(1.467301946 + 0.6528950495i\)
\(L(1)\) \(\approx\) \(1.467301946 + 0.6528950495i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
37 \( 1 \)
good2 \( 1 + (0.642 + 0.766i)T \)
7 \( 1 + (0.939 - 0.342i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (0.984 + 0.173i)T \)
17 \( 1 + (0.984 - 0.173i)T \)
19 \( 1 + (0.642 - 0.766i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (-0.866 + 0.5i)T \)
31 \( 1 + iT \)
41 \( 1 + (0.173 - 0.984i)T \)
43 \( 1 + iT \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (-0.939 - 0.342i)T \)
59 \( 1 + (0.342 - 0.939i)T \)
61 \( 1 + (0.984 + 0.173i)T \)
67 \( 1 + (-0.939 + 0.342i)T \)
71 \( 1 + (-0.766 - 0.642i)T \)
73 \( 1 + T \)
79 \( 1 + (0.342 + 0.939i)T \)
83 \( 1 + (0.173 + 0.984i)T \)
89 \( 1 + (-0.342 + 0.939i)T \)
97 \( 1 + (-0.866 - 0.5i)T \)
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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

−23.094593508931747037644632630863, −22.41550598341400160500604139798, −21.26894841876506226983005078664, −20.747957762754888947713153195741, −20.28686265286287922778063794576, −18.82637142734570895230110984654, −18.50094008109743248235217533422, −17.55202258590250571309765632440, −16.31622274298079175624592363750, −15.13139561163232018510139319889, −14.767463847523121534361419978208, −13.70636942376395975645917069601, −12.88289336938184810920581748001, −12.01608595281874355020847790796, −11.27097860878525956249608111442, −10.39763958579476302330900090918, −9.55172694791851369489310304050, −8.40429561701841057189108753480, −7.42428778192768866147374813341, −5.97541761535034996856564120054, −5.28847315321247134989962934204, −4.327585545303049373478217537611, −3.296115273269017523647068693363, −2.12530297100491565455565576405, −1.212986502421340316411919795526, 1.242268260391196972181342321909, 2.9566537699429806076458629444, 3.757048753868222072975341064511, 5.02625365782696619629621962427, 5.54107490789422215211129227450, 6.75593472886436358224897973306, 7.674389680435587493175070186073, 8.38604383378371713954202571173, 9.32340100628257543637690173763, 10.94505246363261230866577970060, 11.38216819168429142000000330450, 12.59540867781378804884277663891, 13.54400875423711053833016155129, 14.08278896008369903068857542689, 14.94447909285861776618802382272, 15.95299893714819172796036771578, 16.49817941586254570544897281784, 17.55835959812381862070722041320, 18.18039049009149578316481809913, 19.18644197570879646155444056193, 20.65609089980033785796005144632, 21.00976891481838112769480008160, 21.83253841653431077040209824388, 22.845657534906207951295918818078, 23.719778687711794181038069569153

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