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Properties

Label 1-5e3-125.71-r0-0-0
Degree $1$
Conductor $125$
Sign $0.837 - 0.546i$
Analytic cond. $0.580497$
Root an. cond. $0.580497$
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  + (−0.187 − 0.982i)2-s + (0.968 + 0.248i)3-s + (−0.929 + 0.368i)4-s + (0.0627 − 0.998i)6-s + (0.309 + 0.951i)7-s + (0.535 + 0.844i)8-s + (0.876 + 0.481i)9-s + (−0.187 − 0.982i)11-s + (−0.992 + 0.125i)12-s + (0.876 + 0.481i)13-s + (0.876 − 0.481i)14-s + (0.728 − 0.684i)16-s + (−0.929 − 0.368i)17-s + (0.309 − 0.951i)18-s + (0.968 − 0.248i)19-s + ⋯
L(s)  = 1  + (−0.187 − 0.982i)2-s + (0.968 + 0.248i)3-s + (−0.929 + 0.368i)4-s + (0.0627 − 0.998i)6-s + (0.309 + 0.951i)7-s + (0.535 + 0.844i)8-s + (0.876 + 0.481i)9-s + (−0.187 − 0.982i)11-s + (−0.992 + 0.125i)12-s + (0.876 + 0.481i)13-s + (0.876 − 0.481i)14-s + (0.728 − 0.684i)16-s + (−0.929 − 0.368i)17-s + (0.309 − 0.951i)18-s + (0.968 − 0.248i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.837 - 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.837 - 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(125\)    =    \(5^{3}\)
Sign: $0.837 - 0.546i$
Analytic conductor: \(0.580497\)
Root analytic conductor: \(0.580497\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{125} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 125,\ (0:\ ),\ 0.837 - 0.546i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.203214712 - 0.3577794203i\)
\(L(\frac12)\) \(\approx\) \(1.203214712 - 0.3577794203i\)
\(L(1)\) \(\approx\) \(1.168129003 - 0.3194632658i\)
\(L(1)\) \(\approx\) \(1.168129003 - 0.3194632658i\)

Euler product

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

−28.87393035736188734321939255425, −27.60519978402211848269097785468, −26.65805325610998471168299847777, −25.95939461579147948071911632591, −25.107584492330231143204816934774, −24.09403863459618925996307856505, −23.33106394070645925951072819780, −22.14159443997730116435322832773, −20.461024644541768213026235481220, −19.99210297853711321189311626786, −18.41886181631929751428468248369, −17.87407319198976729848059216010, −16.509122013282441638094301348773, −15.4055300081526048081328374072, −14.536857433650215717842833382732, −13.57642588602937943560460700034, −12.76893021033230803939444896136, −10.62157807088858398560618569397, −9.55211443381074120716459358795, −8.354078757704142069721515975099, −7.510969130243581056072322848217, −6.54506318933622060835124001810, −4.75603026959274585079094487271, −3.63222806848687692565661283107, −1.50061470122377835051763885324, 1.773227206281555096149790816259, 2.935958563638710054083672368669, 4.057782157205301318206366645121, 5.55037263447020195196799114782, 7.7356683510063647479956533464, 8.84211543989716694248038041715, 9.369649030780301517974596786342, 10.90901619662245631865150385559, 11.76972496527413480604675776902, 13.30911365626583335963914770646, 13.85852596097355953672155012015, 15.24457954737243794706789806633, 16.32861563587544710802717053136, 18.10537038973876966705984575317, 18.6854177625861324533578485947, 19.722899972667898372739280233517, 20.68047358695926998108644754316, 21.55362440913503797799189387618, 22.17894364520180198173168913347, 23.848241517172211831337166044159, 24.9348468002604698169877689030, 26.125612996843050254649049592693, 26.83379596433008734893301679801, 27.851827750834453909874940060367, 28.743566273825303432420182025779

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