Properties

Degree 1
Conductor $ 5 \cdot 47 $
Sign $0.908 + 0.416i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.631 − 0.775i)2-s + (0.398 + 0.917i)3-s + (−0.203 + 0.979i)4-s + (0.460 − 0.887i)6-s + (0.942 − 0.334i)7-s + (0.887 − 0.460i)8-s + (−0.682 + 0.730i)9-s + (0.576 + 0.816i)11-s + (−0.979 + 0.203i)12-s + (0.997 − 0.0682i)13-s + (−0.854 − 0.519i)14-s + (−0.917 − 0.398i)16-s + (−0.816 − 0.576i)17-s + (0.997 + 0.0682i)18-s + (−0.990 + 0.136i)19-s + ⋯
L(s,χ)  = 1  + (−0.631 − 0.775i)2-s + (0.398 + 0.917i)3-s + (−0.203 + 0.979i)4-s + (0.460 − 0.887i)6-s + (0.942 − 0.334i)7-s + (0.887 − 0.460i)8-s + (−0.682 + 0.730i)9-s + (0.576 + 0.816i)11-s + (−0.979 + 0.203i)12-s + (0.997 − 0.0682i)13-s + (−0.854 − 0.519i)14-s + (−0.917 − 0.398i)16-s + (−0.816 − 0.576i)17-s + (0.997 + 0.0682i)18-s + (−0.990 + 0.136i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.908 + 0.416i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.908 + 0.416i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(235\)    =    \(5 \cdot 47\)
\( \varepsilon \)  =  $0.908 + 0.416i$
motivic weight  =  \(0\)
character  :  $\chi_{235} (52, \cdot )$
Sato-Tate  :  $\mu(92)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 235,\ (0:\ ),\ 0.908 + 0.416i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(1.059790425 + 0.2314496761i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(1.059790425 + 0.2314496761i\)
\(L(\chi,1)\)  \(\approx\)  \(0.9712275889 + 0.05324801297i\)
\(L(1,\chi)\)  \(\approx\)  \(0.9712275889 + 0.05324801297i\)

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

−26.061059789918706281187641154852, −25.18487973159536681112523630108, −24.46890189765371248817872847690, −23.82983641742479199636696772478, −22.97165365066890100616437615596, −21.48046936788372556626995699324, −20.384030671380947501613251603180, −19.18519274023406701830763413955, −18.83362016911096461113438256634, −17.57833876941956533972424481388, −17.24178708509566623356717452818, −15.709196757868675796749842120051, −14.86416001717342873423414457795, −13.95110336979702853332281983791, −13.1739630248555929501218958863, −11.57580080675248193371995593633, −10.85378361409642320292678930711, −9.04961591960775688453141078939, −8.56277200394888358743153297362, −7.683776828070529041609942309708, −6.44644559546557265357174086085, −5.779087876001458754750505729186, −4.120918234301886789105741099485, −2.194011597018744106275781168336, −1.091357225533516872647406225487, 1.52143800125103087630447889137, 2.79486071755484931317502438300, 4.14996851990273401298570900013, 4.74193630240136873172039819929, 6.81509474531283355070812607618, 8.22979625251510212498154576300, 8.81096147490367099853423518952, 9.91407986893475531849987600612, 10.86959257856539358353981200348, 11.45445582184427132680128873517, 12.8617711930474707298177968000, 13.990645792279394378037494090621, 14.96142969553482837706046931782, 16.106734759406609456085426673857, 17.104345557729995437211893730502, 17.84966539296836356812693232709, 18.99460624856402983349736761721, 20.18582619116956332930306838475, 20.55547361691260297539508010200, 21.395980256737836177982158104028, 22.34715281727081212408477011593, 23.27461220734271383037136934144, 24.919784745823605663690483826216, 25.63666841182306174898624693036, 26.61781229727419031093860552733

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