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Properties

Label 1-5e3-125.28-r1-0-0
Degree $1$
Conductor $125$
Sign $0.627 - 0.778i$
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  + (0.684 − 0.728i)2-s + (0.844 + 0.535i)3-s + (−0.0627 − 0.998i)4-s + (0.968 − 0.248i)6-s + (0.951 − 0.309i)7-s + (−0.770 − 0.637i)8-s + (0.425 + 0.904i)9-s + (0.728 + 0.684i)11-s + (0.481 − 0.876i)12-s + (0.904 − 0.425i)13-s + (0.425 − 0.904i)14-s + (−0.992 + 0.125i)16-s + (−0.998 − 0.0627i)17-s + (0.951 + 0.309i)18-s + (−0.535 − 0.844i)19-s + ⋯
L(s)  = 1  + (0.684 − 0.728i)2-s + (0.844 + 0.535i)3-s + (−0.0627 − 0.998i)4-s + (0.968 − 0.248i)6-s + (0.951 − 0.309i)7-s + (−0.770 − 0.637i)8-s + (0.425 + 0.904i)9-s + (0.728 + 0.684i)11-s + (0.481 − 0.876i)12-s + (0.904 − 0.425i)13-s + (0.425 − 0.904i)14-s + (−0.992 + 0.125i)16-s + (−0.998 − 0.0627i)17-s + (0.951 + 0.309i)18-s + (−0.535 − 0.844i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.627 - 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.627 - 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(125\)    =    \(5^{3}\)
Sign: $0.627 - 0.778i$
Analytic conductor: \(13.4331\)
Root analytic conductor: \(13.4331\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{125} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 125,\ (1:\ ),\ 0.627 - 0.778i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.364416071 - 1.609071068i\)
\(L(\frac12)\) \(\approx\) \(3.364416071 - 1.609071068i\)
\(L(1)\) \(\approx\) \(2.082233959 - 0.6921408412i\)
\(L(1)\) \(\approx\) \(2.082233959 - 0.6921408412i\)

Euler product

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

−29.13016495061976959787756339720, −27.35438283686507558263789461417, −26.62468474164462420232736023412, −25.41231000950796001686426540548, −24.773605577868877676733307105525, −23.98290980444549239276845448192, −23.04486784834059947838117829811, −21.498272496847060088193085366076, −21.02407830010577799227685369945, −19.67685114277225624320845760710, −18.40889949217821436420996554202, −17.51377099184892062315137756344, −16.175689814834468287542485661618, −15.01140175733836962888547471915, −14.24232927308458158728597531304, −13.44554062204384049289634031564, −12.26176236764135192462341897551, −11.16785033908318631162205601838, −8.80450691245271953403862040671, −8.48255014226271215724086202238, −7.06062040466016247194365307058, −6.06545440495868650337935691676, −4.45267124263979078439435109556, −3.2766263237739166409171338796, −1.68318313369145099179109723523, 1.423795320374833394111802726792, 2.72854961682504761693169069477, 4.16834120108747644573143633834, 4.82505837488655924511899867310, 6.669343559263845991842525460953, 8.35627073287869558415786094752, 9.41933010808376536192272061993, 10.662103999452982734875411226434, 11.44627415686912198903504070709, 13.04712512119972081159271150819, 13.8330736193268176981019939230, 14.93120783658464129503484346100, 15.48554525395235620894615256262, 17.28376038656129571509675522082, 18.57042703945150320437796096712, 19.828920562362598000220427440855, 20.408348506263237221292465750329, 21.26739558700148396374278671154, 22.20611918957397276414567780403, 23.31539535113448642751743434419, 24.46939322147193584407252751219, 25.347323459637726499516333261560, 26.751977299643126571496077539684, 27.6337332242743177518974804834, 28.360208845343763146422679219689

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