Properties

Label 1-10e2-100.39-r1-0-0
Degree $1$
Conductor $100$
Sign $0.425 - 0.904i$
Analytic cond. $10.7464$
Root an. cond. $10.7464$
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  + (−0.809 − 0.587i)3-s + 7-s + (0.309 + 0.951i)9-s + (−0.309 + 0.951i)11-s + (−0.309 − 0.951i)13-s + (0.809 − 0.587i)17-s + (0.809 − 0.587i)19-s + (−0.809 − 0.587i)21-s + (0.309 − 0.951i)23-s + (0.309 − 0.951i)27-s + (−0.809 − 0.587i)29-s + (0.809 − 0.587i)31-s + (0.809 − 0.587i)33-s + (−0.309 − 0.951i)37-s + (−0.309 + 0.951i)39-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)3-s + 7-s + (0.309 + 0.951i)9-s + (−0.309 + 0.951i)11-s + (−0.309 − 0.951i)13-s + (0.809 − 0.587i)17-s + (0.809 − 0.587i)19-s + (−0.809 − 0.587i)21-s + (0.309 − 0.951i)23-s + (0.309 − 0.951i)27-s + (−0.809 − 0.587i)29-s + (0.809 − 0.587i)31-s + (0.809 − 0.587i)33-s + (−0.309 − 0.951i)37-s + (−0.309 + 0.951i)39-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $0.425 - 0.904i$
Analytic conductor: \(10.7464\)
Root analytic conductor: \(10.7464\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{100} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 100,\ (1:\ ),\ 0.425 - 0.904i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.152259700 - 0.7312462417i\)
\(L(\frac12)\) \(\approx\) \(1.152259700 - 0.7312462417i\)
\(L(1)\) \(\approx\) \(0.9358598968 - 0.2540250827i\)
\(L(1)\) \(\approx\) \(0.9358598968 - 0.2540250827i\)

Euler product

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

−29.62348675175455948657143170016, −28.862864317648190569354303924637, −27.67254527094758985097710054763, −27.03945801957379322856626174956, −25.99040166984662601954589267086, −24.288622393715059497473679419531, −23.77168627725284455223865792646, −22.500746527933434973022986903622, −21.34887291688026555442951841509, −20.93087518459753714318474474348, −19.17987175836405276145918954940, −18.07547485728295299584238691502, −17.00851786843568440386897379672, −16.17547052947023803600766315794, −14.90968176023100092777196599135, −13.850566925811140406548961270076, −12.127258559032661927367061469768, −11.327677058938488326085917184956, −10.28633586044067894083763015481, −8.949389890869736027196310587588, −7.51536704981603626340804836678, −5.903977872380190143192313822978, −4.94659941145312019039306062752, −3.555418614831584361195134171829, −1.298207143180879839028718267543, 0.792213449768409302693341609757, 2.38876140162935409427990490726, 4.690024050922154656117754448364, 5.568335664713866196274496847317, 7.2124093265236748327092897608, 7.975732734127437543419132903096, 9.867910571362079489000546002057, 11.05339551512235533069173483399, 12.05884164505776065298772794622, 13.05380073458289411470742942131, 14.40144086701674480739211980527, 15.61894319877056381564962476881, 17.01505058877383710544906544375, 17.8323258625520663390390987578, 18.592914512003524128651541180257, 20.08890603279608935027598439530, 21.09606106817307240756140497399, 22.51269811540296234566486856716, 23.14420343092907230534774892912, 24.41251454024979104337849073111, 24.97156220988477854066874124064, 26.51888450800937232553276592063, 27.79284901168348543510186134930, 28.29389592309771258936881911290, 29.648336246280855743902313948205

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