Properties

Label 1-10e2-100.47-r0-0-0
Degree $1$
Conductor $100$
Sign $0.998 - 0.0627i$
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  + (−0.951 + 0.309i)3-s i·7-s + (0.809 − 0.587i)9-s + (0.809 + 0.587i)11-s + (0.587 + 0.809i)13-s + (0.951 + 0.309i)17-s + (0.309 − 0.951i)19-s + (0.309 + 0.951i)21-s + (0.587 − 0.809i)23-s + (−0.587 + 0.809i)27-s + (−0.309 − 0.951i)29-s + (−0.309 + 0.951i)31-s + (−0.951 − 0.309i)33-s + (−0.587 − 0.809i)37-s + (−0.809 − 0.587i)39-s + ⋯
L(s)  = 1  + (−0.951 + 0.309i)3-s i·7-s + (0.809 − 0.587i)9-s + (0.809 + 0.587i)11-s + (0.587 + 0.809i)13-s + (0.951 + 0.309i)17-s + (0.309 − 0.951i)19-s + (0.309 + 0.951i)21-s + (0.587 − 0.809i)23-s + (−0.587 + 0.809i)27-s + (−0.309 − 0.951i)29-s + (−0.309 + 0.951i)31-s + (−0.951 − 0.309i)33-s + (−0.587 − 0.809i)37-s + (−0.809 − 0.587i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0627i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $0.998 - 0.0627i$
Analytic conductor: \(0.464398\)
Root analytic conductor: \(0.464398\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{100} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 100,\ (0:\ ),\ 0.998 - 0.0627i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8189813763 + 0.02573752661i\)
\(L(\frac12)\) \(\approx\) \(0.8189813763 + 0.02573752661i\)
\(L(1)\) \(\approx\) \(0.8632529409 + 0.002418697518i\)
\(L(1)\) \(\approx\) \(0.8632529409 + 0.002418697518i\)

Euler product

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

−29.75134497896497924188733929759, −29.03807759897790761052726191491, −27.65833201598717615167126248123, −27.4846744619477263227896054420, −25.49140502196141903299958682400, −24.79873171819603764199974176147, −23.671669920623123485607582635679, −22.58099123726501916654939377274, −21.87194723635468622821665523687, −20.655985623178627664768748627004, −18.99399470145447725385614703782, −18.43068605032943237362972211063, −17.1867108405310791788383110636, −16.231391100036494771766250270246, −15.09471448734655827241744259926, −13.61490649015771129123187723241, −12.331081066506890717813585915630, −11.61840136870746610817257269291, −10.37075448725982881120174553089, −8.93513808453674627269607924228, −7.52648614870437713718767241863, −6.02199089557285058334749582405, −5.3600153803888763914398888020, −3.42761153632103008991521704159, −1.40591797495635540487682333433, 1.252207630791939854476865396509, 3.77373672129255718228815685134, 4.788258772374089186236626011, 6.36649099390229779396272494640, 7.271963046223547151164731628437, 9.17136930469354971096349127851, 10.32154081592810066780959865316, 11.31514049934427781729455966756, 12.38967638140588294963865818397, 13.73700353936956698148899087927, 14.99799374441442619338468919017, 16.38939919655287679779555654895, 17.01686215314292491795645451873, 18.06175006417411582800086336023, 19.3944831300154978376526725295, 20.646948695771474564392637578540, 21.62007648978252239587132095664, 22.85426147720275760530074366890, 23.41869644842605579225625124927, 24.55506554930713135169632655925, 26.03813029430556337126227272641, 26.92201253402264885845502277454, 27.996930739966873129018179952113, 28.75431744763901995096685175974, 29.96850085821985079698546614731

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