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Properties

Label 1-5e3-125.2-r1-0-0
Degree $1$
Conductor $125$
Sign $0.962 + 0.272i$
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  + (0.998 + 0.0627i)2-s + (0.904 + 0.425i)3-s + (0.992 + 0.125i)4-s + (0.876 + 0.481i)6-s + (0.587 − 0.809i)7-s + (0.982 + 0.187i)8-s + (0.637 + 0.770i)9-s + (0.0627 − 0.998i)11-s + (0.844 + 0.535i)12-s + (−0.770 + 0.637i)13-s + (0.637 − 0.770i)14-s + (0.968 + 0.248i)16-s + (−0.125 − 0.992i)17-s + (0.587 + 0.809i)18-s + (0.425 + 0.904i)19-s + ⋯
L(s)  = 1  + (0.998 + 0.0627i)2-s + (0.904 + 0.425i)3-s + (0.992 + 0.125i)4-s + (0.876 + 0.481i)6-s + (0.587 − 0.809i)7-s + (0.982 + 0.187i)8-s + (0.637 + 0.770i)9-s + (0.0627 − 0.998i)11-s + (0.844 + 0.535i)12-s + (−0.770 + 0.637i)13-s + (0.637 − 0.770i)14-s + (0.968 + 0.248i)16-s + (−0.125 − 0.992i)17-s + (0.587 + 0.809i)18-s + (0.425 + 0.904i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.962 + 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.962 + 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(125\)    =    \(5^{3}\)
Sign: $0.962 + 0.272i$
Analytic conductor: \(13.4331\)
Root analytic conductor: \(13.4331\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{125} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 125,\ (1:\ ),\ 0.962 + 0.272i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.658342842 + 0.6480559441i\)
\(L(\frac12)\) \(\approx\) \(4.658342842 + 0.6480559441i\)
\(L(1)\) \(\approx\) \(2.687227291 + 0.2862435232i\)
\(L(1)\) \(\approx\) \(2.687227291 + 0.2862435232i\)

Euler product

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

−28.80615394397019277367777093480, −27.77545780785708651047354318431, −26.208282148099202593291706707249, −25.33904548128474869375412433215, −24.4822849935284831020585059073, −23.8437896486038598613664961427, −22.397703548066447082364690523242, −21.57581871831130951155217008072, −20.39959785577444326065197893772, −19.84667073594169953946779898744, −18.54584865819047009747555811549, −17.363054888721840225406400605157, −15.5583999648675232930657643647, −14.93136474041171287848073280598, −14.16800913439251828521731784319, −12.69356672495693751063408758453, −12.36180735451045480022171453919, −10.786267042843726840953739206555, −9.31407920246429035997864650507, −7.91381039641955719129728272837, −6.949334274539693696957517840056, −5.45418604480892343415318431720, −4.19354755311713529439934875040, −2.6977325997059835120090030481, −1.80088916151307342632635928969, 1.761710739637012397830598668495, 3.2442713477427701930857930886, 4.22300435276784211727279139528, 5.38565059707684506704211352224, 7.14510781506935375602461259409, 7.989406536488315800810189376338, 9.5874580231432283757029425252, 10.846690293940534749782470765621, 11.892353255294519977601458763298, 13.55953591978911208441177949887, 13.96955823408466583236388591289, 14.91212516219889458080581111448, 16.119084968269817173163584326525, 16.91154353297247156119917341071, 18.813245919765653647661075070987, 19.933301745187519357538238576371, 20.68484831921427108207766920204, 21.54621628306319402242001261428, 22.46102929999964415964334447154, 23.87503483354280818628523879066, 24.48209715371570304456209106727, 25.54457166806541432311323679269, 26.6777633315460892836058232963, 27.34173083091853278144830951605, 29.159288739437190733253105782724

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