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Properties

Label 1-4031-4031.3972-r0-0-0
Degree $1$
Conductor $4031$
Sign $0.544 - 0.838i$
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  + (0.576 + 0.816i)2-s + (0.0682 + 0.997i)3-s + (−0.334 + 0.942i)4-s + (0.854 + 0.519i)5-s + (−0.775 + 0.631i)6-s + (−0.775 − 0.631i)7-s + (−0.962 + 0.269i)8-s + (−0.990 + 0.136i)9-s + (0.0682 + 0.997i)10-s + (0.917 + 0.398i)11-s + (−0.962 − 0.269i)12-s + (−0.0682 − 0.997i)13-s + (0.0682 − 0.997i)14-s + (−0.460 + 0.887i)15-s + (−0.775 − 0.631i)16-s + (−0.203 + 0.979i)17-s + ⋯
L(s)  = 1  + (0.576 + 0.816i)2-s + (0.0682 + 0.997i)3-s + (−0.334 + 0.942i)4-s + (0.854 + 0.519i)5-s + (−0.775 + 0.631i)6-s + (−0.775 − 0.631i)7-s + (−0.962 + 0.269i)8-s + (−0.990 + 0.136i)9-s + (0.0682 + 0.997i)10-s + (0.917 + 0.398i)11-s + (−0.962 − 0.269i)12-s + (−0.0682 − 0.997i)13-s + (0.0682 − 0.997i)14-s + (−0.460 + 0.887i)15-s + (−0.775 − 0.631i)16-s + (−0.203 + 0.979i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.544 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.544 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4031\)    =    \(29 \cdot 139\)
Sign: $0.544 - 0.838i$
Analytic conductor: \(18.7198\)
Root analytic conductor: \(18.7198\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4031} (3972, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4031,\ (0:\ ),\ 0.544 - 0.838i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4413312181 + 0.2395527069i\)
\(L(\frac12)\) \(\approx\) \(-0.4413312181 + 0.2395527069i\)
\(L(1)\) \(\approx\) \(0.6874078909 + 0.9206582968i\)
\(L(1)\) \(\approx\) \(0.6874078909 + 0.9206582968i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 \)
139 \( 1 \)
good2 \( 1 + (0.576 + 0.816i)T \)
3 \( 1 + (0.0682 + 0.997i)T \)
5 \( 1 + (0.854 + 0.519i)T \)
7 \( 1 + (-0.775 - 0.631i)T \)
11 \( 1 + (0.917 + 0.398i)T \)
13 \( 1 + (-0.0682 - 0.997i)T \)
17 \( 1 + (-0.203 + 0.979i)T \)
19 \( 1 + (-0.203 + 0.979i)T \)
23 \( 1 + (-0.775 - 0.631i)T \)
31 \( 1 + (0.990 + 0.136i)T \)
37 \( 1 + (-0.460 - 0.887i)T \)
41 \( 1 + (-0.682 + 0.730i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.854 + 0.519i)T \)
53 \( 1 + (-0.576 - 0.816i)T \)
59 \( 1 + (-0.334 - 0.942i)T \)
61 \( 1 + (-0.682 - 0.730i)T \)
67 \( 1 + (-0.917 + 0.398i)T \)
71 \( 1 + (0.854 + 0.519i)T \)
73 \( 1 + (-0.962 + 0.269i)T \)
79 \( 1 + (-0.682 - 0.730i)T \)
83 \( 1 + (-0.917 - 0.398i)T \)
89 \( 1 + (-0.460 + 0.887i)T \)
97 \( 1 - T \)
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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

−18.240134464533113629526017196529, −17.31481154647590073459617601964, −16.7179166480478453287736643167, −15.77781884953941119566764665063, −14.956557001709187161683700707819, −13.9251592788931051910914721159, −13.740864534727170245108833616111, −13.232825787218267706345025524546, −12.34766362157656371553431184395, −11.78663150163938741381649019739, −11.519088069975489236008933279986, −10.25289426080626618667834457830, −9.46415649666488582774311404382, −9.03644574033395003160154512843, −8.49146597972557820673036558744, −6.96553203088403963606095909660, −6.48908206287902585898005491812, −5.93580334568293245718785580270, −5.14834711517848239659638091088, −4.3507626169228363073576500401, −3.2059693092581854741914822495, −2.64013555954201900561353314361, −1.82635575864966158767981981825, −1.260932065575934039336661022165, −0.10249936153184384303271588850, 1.72211707865215984725379647851, 2.89743352305581157862861397696, 3.45279196788370355897394265966, 4.09651871010862729288302845838, 4.851233126755212405345497004715, 5.820486272833254008308071340524, 6.26305913059840921918933238634, 6.812717369011262045069780135193, 7.9396252978123690522472671783, 8.52226038917569116050191497682, 9.49428771989569284387955297416, 10.03607885755774235760333769849, 10.49170844711550021012593498614, 11.49240935776905982149915515886, 12.47951167709552080867796378537, 13.0023244491516148288540973438, 13.891527069378251543524885475591, 14.38274314357106857975377040863, 14.8903996811629298209881370674, 15.56986041242633409574171913450, 16.32450650753134725004867510974, 16.9371429898844507914883842380, 17.378067203772167144011638662, 17.9650192678005620427466939034, 19.05639329999881038738291978677

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