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Properties

Label 1-5e3-125.53-r1-0-0
Degree $1$
Conductor $125$
Sign $-0.546 - 0.837i$
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  + (−0.481 − 0.876i)2-s + (−0.368 − 0.929i)3-s + (−0.535 + 0.844i)4-s + (−0.637 + 0.770i)6-s + (0.951 − 0.309i)7-s + (0.998 + 0.0627i)8-s + (−0.728 + 0.684i)9-s + (0.876 − 0.481i)11-s + (0.982 + 0.187i)12-s + (0.684 + 0.728i)13-s + (−0.728 − 0.684i)14-s + (−0.425 − 0.904i)16-s + (0.844 − 0.535i)17-s + (0.951 + 0.309i)18-s + (0.929 + 0.368i)19-s + ⋯
L(s)  = 1  + (−0.481 − 0.876i)2-s + (−0.368 − 0.929i)3-s + (−0.535 + 0.844i)4-s + (−0.637 + 0.770i)6-s + (0.951 − 0.309i)7-s + (0.998 + 0.0627i)8-s + (−0.728 + 0.684i)9-s + (0.876 − 0.481i)11-s + (0.982 + 0.187i)12-s + (0.684 + 0.728i)13-s + (−0.728 − 0.684i)14-s + (−0.425 − 0.904i)16-s + (0.844 − 0.535i)17-s + (0.951 + 0.309i)18-s + (0.929 + 0.368i)19-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(125\)    =    \(5^{3}\)
Sign: $-0.546 - 0.837i$
Analytic conductor: \(13.4331\)
Root analytic conductor: \(13.4331\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{125} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 125,\ (1:\ ),\ -0.546 - 0.837i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6469732134 - 1.194557879i\)
\(L(\frac12)\) \(\approx\) \(0.6469732134 - 1.194557879i\)
\(L(1)\) \(\approx\) \(0.6992106411 - 0.5874252952i\)
\(L(1)\) \(\approx\) \(0.6992106411 - 0.5874252952i\)

Euler product

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

−28.28039027586046905262324849196, −27.87866329639777650154403070497, −27.1327304851037975111954522122, −26.01745451873779114265340698060, −25.13976126103364406755921704140, −24.02994588508564440322730638850, −22.99131356002570118728631463840, −22.1377308561839250388507060975, −20.89440832080275850209685268368, −19.85487665329143363114192974002, −18.349353170578196285422114289152, −17.4792405901376650635376103817, −16.72962482631377304388256067061, −15.43051122956531910926632958850, −14.93442386711597147451669911319, −13.78886822245487927259478640878, −11.87963070375112075368542365528, −10.794004090015190338851649744473, −9.6700043012990383748290631657, −8.71511736506080702065827202355, −7.515247049365114715606880604708, −5.91585165999884713897619327310, −5.1287129842572052754043514361, −3.79817363459107900865834459774, −1.232496738662715372849352556309, 0.881467666981851218999055407949, 1.80655876471964981689184882461, 3.53421434159851896874147018368, 5.11472951811429790002332363528, 6.81695741007260855236511658644, 7.97728919313039693039369807220, 8.93105494113403019549767589039, 10.53052863628937867148542765700, 11.60537711504560631280139059442, 12.07599508901207239742579793672, 13.64864681232293645225240406339, 14.186289038159001828917402339332, 16.51198221107757155386717875360, 17.1711773506870483577158733564, 18.37499631820458243326481797291, 18.842646544676518979290727294833, 20.1208969889268831408877799370, 20.972957666995007560651394339259, 22.21399676145691206418487896427, 23.169083429244439540317756147717, 24.30947794405168314425513417331, 25.258206848721253027787030794486, 26.57817250815632912111872597903, 27.51126967668537964966538437592, 28.430230961833204997109307369523

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