Properties

Label 1-4031-4031.1897-r0-0-0
Degree $1$
Conductor $4031$
Sign $0.583 - 0.812i$
Analytic cond. $18.7198$
Root an. cond. $18.7198$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(4031\)    =    \(29 \cdot 139\)
Sign: $0.583 - 0.812i$
Analytic conductor: \(18.7198\)
Root analytic conductor: \(18.7198\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4031} (1897, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4031,\ (0:\ ),\ 0.583 - 0.812i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6805693699 - 0.3490465239i\)
\(L(\frac12)\) \(\approx\) \(0.6805693699 - 0.3490465239i\)
\(L(1)\) \(\approx\) \(0.6031780100 + 4.175125791\times10^{-5}i\)
\(L(1)\) \(\approx\) \(0.6031780100 + 4.175125791\times10^{-5}i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 \)
139 \( 1 \)
good2 \( 1 + (-0.987 + 0.158i)T \)
3 \( 1 + (-0.968 + 0.247i)T \)
5 \( 1 + (0.419 - 0.907i)T \)
7 \( 1 + (0.113 + 0.993i)T \)
11 \( 1 + (-0.439 + 0.898i)T \)
13 \( 1 + (0.715 - 0.699i)T \)
17 \( 1 + (-0.225 + 0.974i)T \)
19 \( 1 + (0.956 - 0.291i)T \)
23 \( 1 + (0.917 + 0.398i)T \)
31 \( 1 + (0.480 - 0.877i)T \)
37 \( 1 + (-0.181 - 0.983i)T \)
41 \( 1 + (0.926 - 0.377i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + (0.0909 - 0.995i)T \)
53 \( 1 + (-0.934 - 0.356i)T \)
59 \( 1 + (-0.203 - 0.979i)T \)
61 \( 1 + (-0.926 - 0.377i)T \)
67 \( 1 + (0.829 + 0.557i)T \)
71 \( 1 + (-0.995 + 0.0909i)T \)
73 \( 1 + (0.0455 - 0.998i)T \)
79 \( 1 + (-0.136 - 0.990i)T \)
83 \( 1 + (-0.829 + 0.557i)T \)
89 \( 1 + (-0.761 - 0.648i)T \)
97 \( 1 + (0.866 - 0.5i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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