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Properties

Label 1-4031-4031.17-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

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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: $\chi_{4031} (17, \cdot )$
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 \( 1 + (0.866 + 0.5i)T \)
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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

−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