Properties

Degree 1
Conductor 47
Sign $0.761 + 0.648i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(47\)
\( \varepsilon \)  =  $0.761 + 0.648i$
motivic weight  =  \(0\)
character  :  $\chi_{47} (18, \cdot )$
Sato-Tate  :  $\mu(23)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 47,\ (0:\ ),\ 0.761 + 0.648i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.6326664467 + 0.2327860540i$
$L(\frac12,\chi)$  $\approx$  $0.6326664467 + 0.2327860540i$
$L(\chi,1)$  $\approx$  0.8039000200 + 0.1099745280i
$L(1,\chi)$  $\approx$  0.8039000200 + 0.1099745280i

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

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

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