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Properties

Label 1-5e3-125.48-r1-0-0
Degree $1$
Conductor $125$
Sign $-0.556 + 0.830i$
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  + (0.770 + 0.637i)2-s + (0.125 − 0.992i)3-s + (0.187 + 0.982i)4-s + (0.728 − 0.684i)6-s + (−0.587 + 0.809i)7-s + (−0.481 + 0.876i)8-s + (−0.968 − 0.248i)9-s + (−0.637 + 0.770i)11-s + (0.998 − 0.0627i)12-s + (−0.248 + 0.968i)13-s + (−0.968 + 0.248i)14-s + (−0.929 + 0.368i)16-s + (0.982 + 0.187i)17-s + (−0.587 − 0.809i)18-s + (0.992 − 0.125i)19-s + ⋯
L(s)  = 1  + (0.770 + 0.637i)2-s + (0.125 − 0.992i)3-s + (0.187 + 0.982i)4-s + (0.728 − 0.684i)6-s + (−0.587 + 0.809i)7-s + (−0.481 + 0.876i)8-s + (−0.968 − 0.248i)9-s + (−0.637 + 0.770i)11-s + (0.998 − 0.0627i)12-s + (−0.248 + 0.968i)13-s + (−0.968 + 0.248i)14-s + (−0.929 + 0.368i)16-s + (0.982 + 0.187i)17-s + (−0.587 − 0.809i)18-s + (0.992 − 0.125i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.556 + 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.556 + 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(125\)    =    \(5^{3}\)
Sign: $-0.556 + 0.830i$
Analytic conductor: \(13.4331\)
Root analytic conductor: \(13.4331\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{125} (48, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 125,\ (1:\ ),\ -0.556 + 0.830i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8689736639 + 1.628811206i\)
\(L(\frac12)\) \(\approx\) \(0.8689736639 + 1.628811206i\)
\(L(1)\) \(\approx\) \(1.235458649 + 0.5851716177i\)
\(L(1)\) \(\approx\) \(1.235458649 + 0.5851716177i\)

Euler product

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

−28.43645659012458975129928967731, −27.373721054134768362411622281404, −26.51712889810813820319033030995, −25.32365277534578555091420101244, −23.97350915568183889846346909297, −22.90370696869170735783913565843, −22.258309832177863278023976577729, −21.18161219569144999530394773342, −20.325173619304383870714895315186, −19.62731280858831339665368965439, −18.24620354976427721787846050963, −16.51045583884390541806598322470, −15.86028611986642790417848497770, −14.57875132719180328991233082903, −13.7545777733224296244202958454, −12.61813210147776797635371841404, −11.222381278029153697148180175946, −10.30231221779055070932415960001, −9.625902254229076736045654823750, −7.85274592388478067960334198286, −5.99295645849787226255088787643, −5.01801184678608553693252703524, −3.63931844092062656289047055112, −2.90716143568395887839278812482, −0.546630648604297869495956322617, 2.096072486989182528392383644560, 3.29325413180221771346528741053, 5.12107502510389824176132971868, 6.19127521681722245175778497197, 7.22859509233819854702406511320, 8.21062876003003118752921178460, 9.585074055221436841219514499404, 11.80054040269715053750347834220, 12.28712011051331815093293313023, 13.38493828130058970896521694449, 14.304896500794518724965323425379, 15.427117859575536093465107392, 16.4832941731655151039027317548, 17.725599547409970655709257184669, 18.619019003731317301551788083605, 19.81981605560733611659228929343, 21.075579578829639423298303681256, 22.16834562907706269907293182480, 23.19134039000981700051709612991, 23.91085579342808106110431209764, 24.975419044506352390981394716586, 25.64114765570382262901234246313, 26.4705436154662399556707863555, 28.3351475049182085379496976231, 29.134211788348355416728569231849

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