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Properties

Label 1-10e2-100.23-r0-0-0
Degree $1$
Conductor $100$
Sign $-0.425 + 0.904i$
Analytic cond. $0.464398$
Root an. cond. $0.464398$
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  + (−0.587 + 0.809i)3-s − i·7-s + (−0.309 − 0.951i)9-s + (−0.309 + 0.951i)11-s + (−0.951 + 0.309i)13-s + (0.587 + 0.809i)17-s + (−0.809 + 0.587i)19-s + (−0.809 − 0.587i)21-s + (−0.951 − 0.309i)23-s + (0.951 + 0.309i)27-s + (0.809 + 0.587i)29-s + (0.809 − 0.587i)31-s + (−0.587 − 0.809i)33-s + (0.951 − 0.309i)37-s + (0.309 − 0.951i)39-s + ⋯
L(s)  = 1  + (−0.587 + 0.809i)3-s − i·7-s + (−0.309 − 0.951i)9-s + (−0.309 + 0.951i)11-s + (−0.951 + 0.309i)13-s + (0.587 + 0.809i)17-s + (−0.809 + 0.587i)19-s + (−0.809 − 0.587i)21-s + (−0.951 − 0.309i)23-s + (0.951 + 0.309i)27-s + (0.809 + 0.587i)29-s + (0.809 − 0.587i)31-s + (−0.587 − 0.809i)33-s + (0.951 − 0.309i)37-s + (0.309 − 0.951i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.425 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.425 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $-0.425 + 0.904i$
Analytic conductor: \(0.464398\)
Root analytic conductor: \(0.464398\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{100} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 100,\ (0:\ ),\ -0.425 + 0.904i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3714880144 + 0.5853714437i\)
\(L(\frac12)\) \(\approx\) \(0.3714880144 + 0.5853714437i\)
\(L(1)\) \(\approx\) \(0.6890129110 + 0.3795408780i\)
\(L(1)\) \(\approx\) \(0.6890129110 + 0.3795408780i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-0.587 + 0.809i)T \)
7 \( 1 - iT \)
11 \( 1 + (-0.309 + 0.951i)T \)
13 \( 1 + (-0.951 + 0.309i)T \)
17 \( 1 + (0.587 + 0.809i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (-0.951 - 0.309i)T \)
29 \( 1 + (0.809 + 0.587i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (0.951 - 0.309i)T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.587 - 0.809i)T \)
53 \( 1 + (0.587 - 0.809i)T \)
59 \( 1 + (0.309 + 0.951i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + (-0.587 - 0.809i)T \)
71 \( 1 + (0.809 + 0.587i)T \)
73 \( 1 + (0.951 + 0.309i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + (0.587 + 0.809i)T \)
89 \( 1 + (-0.309 + 0.951i)T \)
97 \( 1 + (-0.587 + 0.809i)T \)
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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

−29.713334882950662559760622048, −28.85623063575398985247955701354, −27.557162418343600732441640636710, −26.62603240151549872140529891033, −25.31445817026874507420028468230, −24.22074452021163789576942599726, −23.50664240343346375970610264874, −22.516172687813663088044346814644, −21.338922534638768811678026058056, −19.87757941094100248029430493207, −19.09721107617207360519891028995, −17.82499688483467679936874681654, −16.99694886756347578591086852968, −15.9945770585903485064452897118, −14.19700529119529658479880287738, −13.412784899242613700657531758493, −12.21623174487602059218225135145, −11.096931425184605939281390124, −10.03463080455948019202641468992, −8.152586563172433997854484732567, −7.2187081970722733749256298332, −5.989880442572423820639942398784, −4.62242380207240665655258369933, −2.70417932007092419010529431626, −0.75954028337927987111452801199, 2.33389632649308245143236151422, 4.14087291560606716122220135327, 5.292376671542625744546087886895, 6.44144232561041201861499977318, 8.20590983890052991989120102855, 9.59179150948552811872569852423, 10.39389239212235687906783824664, 11.93030871380460766631528980681, 12.524010702773494780276491775262, 14.60157398443678000300576471863, 15.23545829756222228881130273640, 16.42986128848997447943828628387, 17.44153649769959241670258326804, 18.50464684410546712474168121690, 19.85731333638669589589568456965, 21.16594852648275237818094756010, 21.828605544469596916031353663505, 22.86145136798583189335821435490, 23.88359075579308150337444196799, 25.245720471434921300250125494471, 26.19687073623897742898748581604, 27.39623093288325115685368584873, 28.20969269308971732198442262816, 28.91615015165956954740309974698, 30.23833429689657663837931835591

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