Properties

Degree 1
Conductor $ 29 \cdot 139 $
Sign $0.583 + 0.812i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.987 − 0.158i)2-s + (−0.968 − 0.247i)3-s + (0.949 + 0.313i)4-s + (0.419 + 0.907i)5-s + (0.917 + 0.398i)6-s + (0.113 − 0.993i)7-s + (−0.887 − 0.460i)8-s + (0.877 + 0.480i)9-s + (−0.269 − 0.962i)10-s + (−0.439 − 0.898i)11-s + (−0.842 − 0.538i)12-s + (0.715 + 0.699i)13-s + (−0.269 + 0.962i)14-s + (−0.181 − 0.983i)15-s + (0.803 + 0.595i)16-s + (−0.225 − 0.974i)17-s + ⋯
L(s,χ)  = 1  + (−0.987 − 0.158i)2-s + (−0.968 − 0.247i)3-s + (0.949 + 0.313i)4-s + (0.419 + 0.907i)5-s + (0.917 + 0.398i)6-s + (0.113 − 0.993i)7-s + (−0.887 − 0.460i)8-s + (0.877 + 0.480i)9-s + (−0.269 − 0.962i)10-s + (−0.439 − 0.898i)11-s + (−0.842 − 0.538i)12-s + (0.715 + 0.699i)13-s + (−0.269 + 0.962i)14-s + (−0.181 − 0.983i)15-s + (0.803 + 0.595i)16-s + (−0.225 − 0.974i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.583 + 0.812i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.583 + 0.812i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4031\)    =    \(29 \cdot 139\)
\( \varepsilon \)  =  $0.583 + 0.812i$
motivic weight  =  \(0\)
character  :  $\chi_{4031} (17, \cdot )$
Sato-Tate  :  $\mu(276)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 4031,\ (0:\ ),\ 0.583 + 0.812i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.6805693699 + 0.3490465239i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.6805693699 + 0.3490465239i\)
\(L(\chi,1)\)  \(\approx\)  \(0.6031780100 + 4.175125791\times10^{-5}i\)
\(L(1,\chi)\)  \(\approx\)  \(0.6031780100 + 4.175125791\times10^{-5}i\)

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

−18.10533952337638369254577887981, −17.54640662447719170131390090740, −17.337862869233759832036087612432, −16.35653917617256639542262314202, −15.70135000467239586723768170196, −15.472662682120381291544721536437, −14.64971488478221708494545864777, −13.281687693697203189947755533925, −12.616338934879538515607132685158, −12.20844321232818474529317259684, −11.2836589779939453508205341834, −10.81286465518101489085549837641, −9.87524207095087657756068381302, −9.45377899088150211241839697941, −8.7694986886673298693113211368, −7.982830234868417254241278674872, −7.250687724462840409455756857231, −6.21817443218666040199633929200, −5.70619157770922353083327116982, −5.23142058133130657836251828369, −4.35026612575595353732623455052, −3.087290435726999045011192133468, −2.01525046042974292584306199932, −1.392126784163724944370612628216, −0.42692236039463223614901548921, 0.97308290522455423103932644258, 1.34296814826721058457100306243, 2.63347898744303254535252662043, 3.23636878703628396236540602793, 4.29934557203974479722942858989, 5.363016179192863523190183904148, 6.20292431953983304653389187779, 6.718526190615205872584625356347, 7.33083682976442223837391861008, 7.90363017954009403568989371679, 9.003742004030786365572930249583, 9.746493499188727233025994815172, 10.47433240789559137206498839987, 10.99001806261185504372080842602, 11.30659170282949668824216661914, 12.0834469968693426379173632592, 13.12884472650102639382100434583, 13.775712911470017984472464986993, 14.33258453786711328981568069960, 15.67469670687570665945224301, 16.00321892842182858758009838458, 16.73467679423913547759674248798, 17.28674836516557510541590790579, 17.97515191658977328940958155593, 18.51339397825501596968813736418

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