Properties

Label 1-47-47.34-r0-0-0
Degree $1$
Conductor $47$
Sign $0.761 - 0.648i$
Analytic cond. $0.218267$
Root an. cond. $0.218267$
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  + (−0.334 + 0.942i)2-s + (0.203 − 0.979i)3-s + (−0.775 − 0.631i)4-s + (−0.0682 − 0.997i)5-s + (0.854 + 0.519i)6-s + (−0.576 − 0.816i)7-s + (0.854 − 0.519i)8-s + (−0.917 − 0.398i)9-s + (0.962 + 0.269i)10-s + (0.460 + 0.887i)11-s + (−0.775 + 0.631i)12-s + (0.682 + 0.730i)13-s + (0.962 − 0.269i)14-s + (−0.990 − 0.136i)15-s + (0.203 + 0.979i)16-s + (0.460 − 0.887i)17-s + ⋯
L(s)  = 1  + (−0.334 + 0.942i)2-s + (0.203 − 0.979i)3-s + (−0.775 − 0.631i)4-s + (−0.0682 − 0.997i)5-s + (0.854 + 0.519i)6-s + (−0.576 − 0.816i)7-s + (0.854 − 0.519i)8-s + (−0.917 − 0.398i)9-s + (0.962 + 0.269i)10-s + (0.460 + 0.887i)11-s + (−0.775 + 0.631i)12-s + (0.682 + 0.730i)13-s + (0.962 − 0.269i)14-s + (−0.990 − 0.136i)15-s + (0.203 + 0.979i)16-s + (0.460 − 0.887i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.761 - 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.761 - 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(47\)
Sign: $0.761 - 0.648i$
Analytic conductor: \(0.218267\)
Root analytic conductor: \(0.218267\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{47} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 47,\ (0:\ ),\ 0.761 - 0.648i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6326664467 - 0.2327860540i\)
\(L(\frac12)\) \(\approx\) \(0.6326664467 - 0.2327860540i\)
\(L(1)\) \(\approx\) \(0.8039000200 - 0.1099745280i\)
\(L(1)\) \(\approx\) \(0.8039000200 - 0.1099745280i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad47 \( 1 \)
good2 \( 1 + (-0.334 + 0.942i)T \)
3 \( 1 + (0.203 - 0.979i)T \)
5 \( 1 + (-0.0682 - 0.997i)T \)
7 \( 1 + (-0.576 - 0.816i)T \)
11 \( 1 + (0.460 + 0.887i)T \)
13 \( 1 + (0.682 + 0.730i)T \)
17 \( 1 + (0.460 - 0.887i)T \)
19 \( 1 + (-0.0682 + 0.997i)T \)
23 \( 1 + (-0.334 - 0.942i)T \)
29 \( 1 + (0.682 - 0.730i)T \)
31 \( 1 + (0.203 + 0.979i)T \)
37 \( 1 + (0.962 + 0.269i)T \)
41 \( 1 + (0.854 + 0.519i)T \)
43 \( 1 + (-0.775 - 0.631i)T \)
53 \( 1 + (0.854 + 0.519i)T \)
59 \( 1 + (-0.775 + 0.631i)T \)
61 \( 1 + (0.962 - 0.269i)T \)
67 \( 1 + (-0.576 + 0.816i)T \)
71 \( 1 + (-0.334 - 0.942i)T \)
73 \( 1 + (-0.917 + 0.398i)T \)
79 \( 1 + (-0.990 - 0.136i)T \)
83 \( 1 + (0.460 + 0.887i)T \)
89 \( 1 + (-0.0682 - 0.997i)T \)
97 \( 1 + (0.203 - 0.979i)T \)
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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

−34.51147465094972359748732368144, −32.760781229692999486521836283736, −31.82188912910045191177205320276, −30.67744644315947629632643606160, −29.6057184276464476523169661561, −28.213437616076975584498343396585, −27.44365820290732572973960408404, −26.235908868856665585487369229425, −25.54936302788395545081177093726, −23.05489785182110166614386788059, −21.93335221538390019009129126981, −21.524878731533080764571581360157, −19.839916443687986850117100351903, −19.00284855262403129670040457427, −17.67172817403053854640932957469, −16.08952829077632371852024274669, −14.81139094347824581923353911122, −13.38966660323227343857106426805, −11.58744958270362759188400866611, −10.65406162845602854076056350583, −9.478507945659891378043997964799, −8.27614044477031593576035137565, −5.84249747968992863564941298874, −3.70600184705884017134004188185, −2.80478196950694983891548706542, 1.20958362185050350126488009537, 4.307541438980147409695869141810, 6.17399511417007680228406114264, 7.32270987816231367064136319505, 8.556454199789278781152701148754, 9.80989198606172332452812591692, 12.16302091601027197968670170443, 13.38427642439584059080447994407, 14.341286060551002932114692037118, 16.19425373473041721418838056921, 17.00282529797616187990911749883, 18.26108788796643636208692210024, 19.503011553064567739137823929643, 20.50810451223241721895781782846, 22.96662832566682865357390626328, 23.49152117146144436114052541581, 24.81287926721951438592547610386, 25.44860144795558328474840987375, 26.74481722940798595212137337263, 28.209679908163127429109436440071, 29.111929773805533412141216104794, 30.69019539368123080176358870600, 31.85885384513593082018709513865, 32.82792716665669938715484134687, 34.01216453273564407411225377556

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