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Properties

Label 1-4031-4031.157-r0-0-0
Degree $1$
Conductor $4031$
Sign $0.0269 - 0.999i$
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  + (0.595 − 0.803i)2-s + (0.842 − 0.538i)3-s + (−0.291 − 0.956i)4-s + (0.158 + 0.987i)5-s + (0.0682 − 0.997i)6-s + (0.898 + 0.439i)7-s + (−0.942 − 0.334i)8-s + (0.419 − 0.907i)9-s + (0.887 + 0.460i)10-s + (0.968 − 0.247i)11-s + (−0.761 − 0.648i)12-s + (0.998 − 0.0455i)13-s + (0.887 − 0.460i)14-s + (0.665 + 0.746i)15-s + (−0.829 + 0.557i)16-s + (−0.789 − 0.613i)17-s + ⋯
L(s)  = 1  + (0.595 − 0.803i)2-s + (0.842 − 0.538i)3-s + (−0.291 − 0.956i)4-s + (0.158 + 0.987i)5-s + (0.0682 − 0.997i)6-s + (0.898 + 0.439i)7-s + (−0.942 − 0.334i)8-s + (0.419 − 0.907i)9-s + (0.887 + 0.460i)10-s + (0.968 − 0.247i)11-s + (−0.761 − 0.648i)12-s + (0.998 − 0.0455i)13-s + (0.887 − 0.460i)14-s + (0.665 + 0.746i)15-s + (−0.829 + 0.557i)16-s + (−0.789 − 0.613i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0269 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0269 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4031\)    =    \(29 \cdot 139\)
Sign: $0.0269 - 0.999i$
Analytic conductor: \(18.7198\)
Root analytic conductor: \(18.7198\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4031} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4031,\ (0:\ ),\ 0.0269 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.152136116 - 3.068237596i\)
\(L(\frac12)\) \(\approx\) \(3.152136116 - 3.068237596i\)
\(L(1)\) \(\approx\) \(1.954696148 - 1.172688536i\)
\(L(1)\) \(\approx\) \(1.954696148 - 1.172688536i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 \)
139 \( 1 \)
good2 \( 1 + (0.595 - 0.803i)T \)
3 \( 1 + (0.842 - 0.538i)T \)
5 \( 1 + (0.158 + 0.987i)T \)
7 \( 1 + (0.898 + 0.439i)T \)
11 \( 1 + (0.968 - 0.247i)T \)
13 \( 1 + (0.998 - 0.0455i)T \)
17 \( 1 + (-0.789 - 0.613i)T \)
19 \( 1 + (0.926 - 0.377i)T \)
23 \( 1 + (0.0682 + 0.997i)T \)
31 \( 1 + (0.907 - 0.419i)T \)
37 \( 1 + (0.665 - 0.746i)T \)
41 \( 1 + (0.999 - 0.0227i)T \)
43 \( 1 + (-0.866 + 0.5i)T \)
47 \( 1 + (-0.356 - 0.934i)T \)
53 \( 1 + (-0.113 + 0.993i)T \)
59 \( 1 + (-0.682 - 0.730i)T \)
61 \( 1 + (-0.999 - 0.0227i)T \)
67 \( 1 + (0.715 + 0.699i)T \)
71 \( 1 + (-0.934 - 0.356i)T \)
73 \( 1 + (0.181 + 0.983i)T \)
79 \( 1 + (-0.519 + 0.854i)T \)
83 \( 1 + (-0.715 + 0.699i)T \)
89 \( 1 + (0.313 - 0.949i)T \)
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

−18.44037613997343458649969769685, −17.71759625543654562209286224784, −17.06067168611018377958000180919, −16.44617804209716360443449801207, −15.92446047343204766345062257299, −15.15055189779032690887937318052, −14.564086824021486255792135784197, −13.88461593449228119304882787879, −13.49880396101775926265772794528, −12.74500339367649681798406351288, −11.91116097857459385477266025043, −11.19472357673103283604854743612, −10.21726190370060652388215436938, −9.27797831616568826936083000370, −8.74608172914157799990376217672, −8.21900285453048748596027134525, −7.66473797350403451852323810173, −6.62793960025684073574002284604, −5.91696400719993173112096958990, −4.85896207800158246823510437981, −4.47592742779177185339920320482, −3.94465792914975435097649390087, −3.08970927666076348440295065057, −1.90289225086059542405340378444, −1.16529626507062394081407300309, 1.02874882981483691577539556580, 1.63898683366425418281066393377, 2.47181747003806127379901048778, 3.063025575062490806113646347067, 3.7914936819590044622058080735, 4.51164993995425556332261482352, 5.661723832814325832613322762339, 6.25927230876853459408839925747, 7.00493400976625803034142699828, 7.820354431592910061409422525654, 8.77765373989996453803299687224, 9.3027644885527624594018123921, 9.96498805523539387931264982185, 11.18945784845735201572221766809, 11.34537192091042819478850416926, 11.97079246924176666185544692347, 13.018110385148644139245793892325, 13.71778264265798561815491863433, 14.02725323137288061348721998403, 14.617612399134644307704685542079, 15.390152788633425433643111692061, 15.732123513980926263044433041701, 17.36128109645665684651173027726, 18.06099240907468880399954666867, 18.35681142460569288389599280497

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