Properties

Label 1-95-95.92-r1-0-0
Degree 11
Conductor 9595
Sign 0.0451+0.998i-0.0451 + 0.998i
Analytic cond. 10.209110.2091
Root an. cond. 10.209110.2091
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: 11
Conductor: 9595    =    5195 \cdot 19
Sign: 0.0451+0.998i-0.0451 + 0.998i
Analytic conductor: 10.209110.2091
Root analytic conductor: 10.209110.2091
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ95(92,)\chi_{95} (92, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 95, (1: ), 0.0451+0.998i)(1,\ 95,\ (1:\ ),\ -0.0451 + 0.998i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.1436520836+0.1502856956i0.1436520836 + 0.1502856956i
L(12)L(\frac12) \approx 0.1436520836+0.1502856956i0.1436520836 + 0.1502856956i
L(1)L(1) \approx 0.42825532840.07429059458i0.4282553284 - 0.07429059458i
L(1)L(1) \approx 0.42825532840.07429059458i0.4282553284 - 0.07429059458i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad5 1 1
19 1 1
good2 1+(0.984+0.173i)T 1 + (-0.984 + 0.173i)T
3 1+(0.6420.766i)T 1 + (-0.642 - 0.766i)T
7 1+(0.8660.5i)T 1 + (-0.866 - 0.5i)T
11 1+(0.50.866i)T 1 + (-0.5 - 0.866i)T
13 1+(0.6420.766i)T 1 + (0.642 - 0.766i)T
17 1+(0.984+0.173i)T 1 + (-0.984 + 0.173i)T
23 1+(0.342+0.939i)T 1 + (0.342 + 0.939i)T
29 1+(0.173+0.984i)T 1 + (-0.173 + 0.984i)T
31 1+(0.5+0.866i)T 1 + (-0.5 + 0.866i)T
37 1+iT 1 + iT
41 1+(0.7660.642i)T 1 + (0.766 - 0.642i)T
43 1+(0.342+0.939i)T 1 + (-0.342 + 0.939i)T
47 1+(0.984+0.173i)T 1 + (0.984 + 0.173i)T
53 1+(0.342+0.939i)T 1 + (0.342 + 0.939i)T
59 1+(0.1730.984i)T 1 + (-0.173 - 0.984i)T
61 1+(0.939+0.342i)T 1 + (-0.939 + 0.342i)T
67 1+(0.984+0.173i)T 1 + (0.984 + 0.173i)T
71 1+(0.9390.342i)T 1 + (-0.939 - 0.342i)T
73 1+(0.6420.766i)T 1 + (-0.642 - 0.766i)T
79 1+(0.766+0.642i)T 1 + (-0.766 + 0.642i)T
83 1+(0.866+0.5i)T 1 + (0.866 + 0.5i)T
89 1+(0.7660.642i)T 1 + (-0.766 - 0.642i)T
97 1+(0.984+0.173i)T 1 + (-0.984 + 0.173i)T
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   L(s)=p (1αpps)1L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}

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 ZZ-function along the critical line