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Properties

Label 1-5e3-125.109-r0-0-0
Degree $1$
Conductor $125$
Sign $0.0376 - 0.999i$
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  + (−0.968 − 0.248i)2-s + (0.187 − 0.982i)3-s + (0.876 + 0.481i)4-s + (−0.425 + 0.904i)6-s + (0.809 − 0.587i)7-s + (−0.728 − 0.684i)8-s + (−0.929 − 0.368i)9-s + (0.968 + 0.248i)11-s + (0.637 − 0.770i)12-s + (0.929 + 0.368i)13-s + (−0.929 + 0.368i)14-s + (0.535 + 0.844i)16-s + (−0.876 + 0.481i)17-s + (0.809 + 0.587i)18-s + (−0.187 − 0.982i)19-s + ⋯
L(s)  = 1  + (−0.968 − 0.248i)2-s + (0.187 − 0.982i)3-s + (0.876 + 0.481i)4-s + (−0.425 + 0.904i)6-s + (0.809 − 0.587i)7-s + (−0.728 − 0.684i)8-s + (−0.929 − 0.368i)9-s + (0.968 + 0.248i)11-s + (0.637 − 0.770i)12-s + (0.929 + 0.368i)13-s + (−0.929 + 0.368i)14-s + (0.535 + 0.844i)16-s + (−0.876 + 0.481i)17-s + (0.809 + 0.587i)18-s + (−0.187 − 0.982i)19-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(125\)    =    \(5^{3}\)
Sign: $0.0376 - 0.999i$
Analytic conductor: \(0.580497\)
Root analytic conductor: \(0.580497\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{125} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 125,\ (0:\ ),\ 0.0376 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5800014860 - 0.5585379805i\)
\(L(\frac12)\) \(\approx\) \(0.5800014860 - 0.5585379805i\)
\(L(1)\) \(\approx\) \(0.7195000760 - 0.3770903922i\)
\(L(1)\) \(\approx\) \(0.7195000760 - 0.3770903922i\)

Euler product

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

−28.7313807721876165048212204972, −27.695309591631302528645303539037, −27.39750585469135354915464898375, −26.30438885421279203832464605968, −25.24014204359884808908324491225, −24.59425621202644557649479810125, −23.11620914005538625050109437627, −21.8189316565600370265659013006, −20.83720047500129412781608543638, −20.06373704134383177561679862815, −18.859053418749742477251006308794, −17.74506583726066342041071717146, −16.82241485878210047934434378547, −15.726278291975088772240288068114, −15.00795680979102884410933030581, −13.937007908298705843619690815686, −11.71332031426779439237219493392, −11.08940688588847589348763696136, −9.82852637393706053166945373081, −8.81467521453512106091538903612, −8.10801868720187210602243933832, −6.33211790797138857439283258236, −5.17366362930862587715682292020, −3.48890881338880187135395510353, −1.783830665687427562860364951619, 1.13834277599394963755167123437, 2.23687908462478218108429392977, 4.01546306782573053426435142367, 6.32564908588060664771406141682, 7.1443249951268014330887024660, 8.34456792324063346462535806832, 9.11151380340438893011285405206, 10.86866890918474065868564774388, 11.53665900884484086520066629777, 12.78488003022982242324964548797, 13.97714590735195159742902369194, 15.19279187970820874502631171786, 16.79085096351829319053020727596, 17.541245245936542822541601822886, 18.34581374659584650607387990772, 19.464483890536880578530408106162, 20.180737924648074986260038703683, 21.15642479974307291916143336822, 22.69174915517340234181757699825, 24.144830733716639507082675776355, 24.543465091088740869332971307836, 25.8664849838099974527517183499, 26.48881625471841337234725376715, 27.85255652740273613437858800257, 28.51494160608379255951531414423

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