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Properties

Label 1-95-95.63-r1-0-0
Degree $1$
Conductor $95$
Sign $-0.0451 - 0.998i$
Analytic cond. $10.2091$
Root an. cond. $10.2091$
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  + (−0.984 − 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

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

Invariants

Degree: \(1\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $-0.0451 - 0.998i$
Analytic conductor: \(10.2091\)
Root analytic conductor: \(10.2091\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 95,\ (1:\ ),\ -0.0451 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1436520836 - 0.1502856956i\)
\(L(\frac12)\) \(\approx\) \(0.1436520836 - 0.1502856956i\)
\(L(1)\) \(\approx\) \(0.4282553284 + 0.07429059458i\)
\(L(1)\) \(\approx\) \(0.4282553284 + 0.07429059458i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.984 - 0.173i)T \)
3 \( 1 + (-0.642 + 0.766i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.642 + 0.766i)T \)
17 \( 1 + (-0.984 - 0.173i)T \)
23 \( 1 + (0.342 - 0.939i)T \)
29 \( 1 + (-0.173 - 0.984i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 - iT \)
41 \( 1 + (0.766 + 0.642i)T \)
43 \( 1 + (-0.342 - 0.939i)T \)
47 \( 1 + (0.984 - 0.173i)T \)
53 \( 1 + (0.342 - 0.939i)T \)
59 \( 1 + (-0.173 + 0.984i)T \)
61 \( 1 + (-0.939 - 0.342i)T \)
67 \( 1 + (0.984 - 0.173i)T \)
71 \( 1 + (-0.939 + 0.342i)T \)
73 \( 1 + (-0.642 + 0.766i)T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (0.866 - 0.5i)T \)
89 \( 1 + (-0.766 + 0.642i)T \)
97 \( 1 + (-0.984 - 0.173i)T \)
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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

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

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