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Properties

Label 1-5e3-125.23-r1-0-0
Degree $1$
Conductor $125$
Sign $-0.0376 - 0.999i$
Analytic cond. $13.4331$
Root an. cond. $13.4331$
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  + (−0.368 + 0.929i)2-s + (0.481 + 0.876i)3-s + (−0.728 − 0.684i)4-s + (−0.992 + 0.125i)6-s + (−0.587 + 0.809i)7-s + (0.904 − 0.425i)8-s + (−0.535 + 0.844i)9-s + (−0.929 − 0.368i)11-s + (0.248 − 0.968i)12-s + (0.844 + 0.535i)13-s + (−0.535 − 0.844i)14-s + (0.0627 + 0.998i)16-s + (−0.684 − 0.728i)17-s + (−0.587 − 0.809i)18-s + (−0.876 − 0.481i)19-s + ⋯
L(s)  = 1  + (−0.368 + 0.929i)2-s + (0.481 + 0.876i)3-s + (−0.728 − 0.684i)4-s + (−0.992 + 0.125i)6-s + (−0.587 + 0.809i)7-s + (0.904 − 0.425i)8-s + (−0.535 + 0.844i)9-s + (−0.929 − 0.368i)11-s + (0.248 − 0.968i)12-s + (0.844 + 0.535i)13-s + (−0.535 − 0.844i)14-s + (0.0627 + 0.998i)16-s + (−0.684 − 0.728i)17-s + (−0.587 − 0.809i)18-s + (−0.876 − 0.481i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0376 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0376 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(125\)    =    \(5^{3}\)
Sign: $-0.0376 - 0.999i$
Analytic conductor: \(13.4331\)
Root analytic conductor: \(13.4331\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{125} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 125,\ (1:\ ),\ -0.0376 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2141665819 + 0.2223965784i\)
\(L(\frac12)\) \(\approx\) \(-0.2141665819 + 0.2223965784i\)
\(L(1)\) \(\approx\) \(0.4776961066 + 0.4927765810i\)
\(L(1)\) \(\approx\) \(0.4776961066 + 0.4927765810i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
good2 \( 1 + (-0.368 + 0.929i)T \)
3 \( 1 + (0.481 + 0.876i)T \)
7 \( 1 + (-0.587 + 0.809i)T \)
11 \( 1 + (-0.929 - 0.368i)T \)
13 \( 1 + (0.844 + 0.535i)T \)
17 \( 1 + (-0.684 - 0.728i)T \)
19 \( 1 + (-0.876 - 0.481i)T \)
23 \( 1 + (-0.770 - 0.637i)T \)
29 \( 1 + (0.187 + 0.982i)T \)
31 \( 1 + (0.728 - 0.684i)T \)
37 \( 1 + (0.998 - 0.0627i)T \)
41 \( 1 + (-0.637 - 0.770i)T \)
43 \( 1 + (-0.951 + 0.309i)T \)
47 \( 1 + (0.904 + 0.425i)T \)
53 \( 1 + (0.125 - 0.992i)T \)
59 \( 1 + (-0.968 - 0.248i)T \)
61 \( 1 + (-0.637 + 0.770i)T \)
67 \( 1 + (0.982 + 0.187i)T \)
71 \( 1 + (-0.425 + 0.904i)T \)
73 \( 1 + (0.248 + 0.968i)T \)
79 \( 1 + (-0.876 + 0.481i)T \)
83 \( 1 + (-0.481 + 0.876i)T \)
89 \( 1 + (-0.968 + 0.248i)T \)
97 \( 1 + (-0.982 + 0.187i)T \)
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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

−28.25993265198414721676893384770, −26.75596755407406421646635541097, −26.02930574807978741042174693755, −25.23334811897364088686194599165, −23.47004788341533996556866638999, −23.1278471927168854157739243767, −21.54007446107880133807595143356, −20.40756867447967614085644510571, −19.82201042708230106485364097877, −18.80355153546716993490408461175, −17.92450488171079549050674810787, −16.98487446002545126557321691667, −15.39243610246169263808515072511, −13.70147761600001844801046884929, −13.19601131233804274751831433138, −12.26381225581774328374821477368, −10.83178342001671516407803157663, −9.90472926312558284046493108511, −8.46243014242893425613433080480, −7.69835946770976764425596551180, −6.23905599122267222403364875378, −4.134205840574753987719120615611, −2.95826777714098901208895174685, −1.62223655339500332859729070611, −0.129958226768775799003712610108, 2.5537241572236598113694796440, 4.22393279946940768906209815677, 5.43368783795330744571140877779, 6.5856878306660748586213581558, 8.27227003384975158550989745040, 8.916236295697560239922401764988, 9.979739994518377689630800580988, 11.10681401699957401773553110676, 13.08983525214475372539134590361, 14.02989732744486262342059183658, 15.29458621695231944397335010162, 15.86364041276329438523379890255, 16.65506151581780482133304666859, 18.20305896930273280306809192263, 18.97237656244383986528401423443, 20.16982818324770216950652646731, 21.482524661324071938752886371393, 22.324133976128190110189738348883, 23.45594586894437226883141958943, 24.61901286946988334101860675959, 25.73281812851152306075196578039, 26.12889714164129209526283841861, 27.191801429445058628647497930824, 28.22577328150852261374794703262, 28.84659021047983224142774138735

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