Properties

Degree 1
Conductor $ 5 \cdot 19 $
Sign $-0.0451 + 0.998i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.984 + 0.173i)2-s + (−0.642 − 0.766i)3-s + (0.939 − 0.342i)4-s + (0.766 + 0.642i)6-s + (−0.866 − 0.5i)7-s + (−0.866 + 0.5i)8-s + (−0.173 + 0.984i)9-s + (−0.5 − 0.866i)11-s + (−0.866 − 0.5i)12-s + (0.642 − 0.766i)13-s + (0.939 + 0.342i)14-s + (0.766 − 0.642i)16-s + (−0.984 + 0.173i)17-s i·18-s + (0.173 + 0.984i)21-s + (0.642 + 0.766i)22-s + ⋯
L(s,χ)  = 1  + (−0.984 + 0.173i)2-s + (−0.642 − 0.766i)3-s + (0.939 − 0.342i)4-s + (0.766 + 0.642i)6-s + (−0.866 − 0.5i)7-s + (−0.866 + 0.5i)8-s + (−0.173 + 0.984i)9-s + (−0.5 − 0.866i)11-s + (−0.866 − 0.5i)12-s + (0.642 − 0.766i)13-s + (0.939 + 0.342i)14-s + (0.766 − 0.642i)16-s + (−0.984 + 0.173i)17-s i·18-s + (0.173 + 0.984i)21-s + (0.642 + 0.766i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.0451 + 0.998i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.0451 + 0.998i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(95\)    =    \(5 \cdot 19\)
\( \varepsilon \)  =  $-0.0451 + 0.998i$
motivic weight  =  \(0\)
character  :  $\chi_{95} (92, \cdot )$
Sato-Tate  :  $\mu(36)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 95,\ (1:\ ),\ -0.0451 + 0.998i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.1436520836 + 0.1502856956i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.1436520836 + 0.1502856956i\)
\(L(\chi,1)\)  \(\approx\)  \(0.4282553284 - 0.07429059458i\)
\(L(1,\chi)\)  \(\approx\)  \(0.4282553284 - 0.07429059458i\)

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−28.98207773829777556462098702207, −28.64620442062038688021277548182, −27.79473234980270382714075231076, −26.46597324209583734090848817216, −26.001659616700418680822838293722, −24.68672870966788838053691453419, −23.233670971268265299863330672420, −22.16059673645781580087053050218, −21.07716623454390098936151145511, −20.17532364676240719416143179907, −18.81161269493234259520004378751, −17.91540886093386327651935445586, −16.75248617732612879162796910551, −15.88786249077670719304804100951, −15.10830166560752309934109080180, −12.887228404319183504174151722832, −11.75955261472416264979917393433, −10.70369060064943568160829002882, −9.63116013788155677338138631695, −8.83377921553808442683990748986, −6.98453444571699646198474453849, −5.93498314677774285899199666290, −4.10282439469411230378226608506, −2.41258825556532452438266435794, −0.15518387887657380654473755102, 1.16225815372873234039051529892, 3.01812984420647006410439813981, 5.59641487452379385536890181018, 6.58408259083114614173576690756, 7.65157725652973523181411817443, 8.8805329070016079178899259681, 10.49446631512216074692268867514, 11.14330033557221085818200844883, 12.66743581375039627476725981991, 13.67297813569562799299730140174, 15.61667657476936079905809927336, 16.42722451519169957063327377927, 17.48034766461596565548568706886, 18.39389439070306364600676040959, 19.34335187342176277943929746003, 20.22214943951463734361671605533, 21.84342577904824549139746254562, 23.205890710466354952890694488867, 23.985468045312825624233939621947, 25.1188035496286303116888543477, 26.00084504398287557169846927817, 27.146122014673284149389232200670, 28.22306528457869866498943386305, 29.26158723496086771981993327484, 29.6037496225037409061991213510

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