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Properties

Label 1-4031-4031.36-r0-0-0
Degree $1$
Conductor $4031$
Sign $0.943 + 0.331i$
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  + (0.972 + 0.232i)2-s + (0.811 + 0.584i)3-s + (0.892 + 0.451i)4-s + (0.165 − 0.986i)5-s + (0.653 + 0.756i)6-s + (−0.883 − 0.468i)7-s + (0.763 + 0.646i)8-s + (0.316 + 0.948i)9-s + (0.389 − 0.921i)10-s + (0.987 − 0.155i)11-s + (0.460 + 0.887i)12-s + (−0.477 − 0.878i)13-s + (−0.750 − 0.660i)14-s + (0.710 − 0.703i)15-s + (0.592 + 0.805i)16-s + (0.682 + 0.730i)17-s + ⋯
L(s)  = 1  + (0.972 + 0.232i)2-s + (0.811 + 0.584i)3-s + (0.892 + 0.451i)4-s + (0.165 − 0.986i)5-s + (0.653 + 0.756i)6-s + (−0.883 − 0.468i)7-s + (0.763 + 0.646i)8-s + (0.316 + 0.948i)9-s + (0.389 − 0.921i)10-s + (0.987 − 0.155i)11-s + (0.460 + 0.887i)12-s + (−0.477 − 0.878i)13-s + (−0.750 − 0.660i)14-s + (0.710 − 0.703i)15-s + (0.592 + 0.805i)16-s + (0.682 + 0.730i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.943 + 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.943 + 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4031\)    =    \(29 \cdot 139\)
Sign: $0.943 + 0.331i$
Analytic conductor: \(18.7198\)
Root analytic conductor: \(18.7198\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4031} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4031,\ (0:\ ),\ 0.943 + 0.331i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.032504651 + 0.8580923002i\)
\(L(\frac12)\) \(\approx\) \(5.032504651 + 0.8580923002i\)
\(L(1)\) \(\approx\) \(2.601747245 + 0.4077578728i\)
\(L(1)\) \(\approx\) \(2.601747245 + 0.4077578728i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 \)
139 \( 1 \)
good2 \( 1 + (0.972 + 0.232i)T \)
3 \( 1 + (0.811 + 0.584i)T \)
5 \( 1 + (0.165 - 0.986i)T \)
7 \( 1 + (-0.883 - 0.468i)T \)
11 \( 1 + (0.987 - 0.155i)T \)
13 \( 1 + (-0.477 - 0.878i)T \)
17 \( 1 + (0.682 + 0.730i)T \)
19 \( 1 + (0.996 - 0.0779i)T \)
23 \( 1 + (0.999 + 0.0390i)T \)
31 \( 1 + (-0.995 - 0.0974i)T \)
37 \( 1 + (0.993 - 0.116i)T \)
41 \( 1 + (-0.990 + 0.136i)T \)
43 \( 1 + (-0.900 - 0.433i)T \)
47 \( 1 + (-0.668 + 0.744i)T \)
53 \( 1 + (0.972 + 0.232i)T \)
59 \( 1 + (0.203 - 0.979i)T \)
61 \( 1 + (-0.511 - 0.859i)T \)
67 \( 1 + (0.987 + 0.155i)T \)
71 \( 1 + (0.924 + 0.380i)T \)
73 \( 1 + (-0.799 + 0.600i)T \)
79 \( 1 + (0.353 - 0.935i)T \)
83 \( 1 + (0.737 + 0.675i)T \)
89 \( 1 + (-0.107 - 0.994i)T \)
97 \( 1 + (0.623 - 0.781i)T \)
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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

−18.53621456282001831493568969251, −18.21475333660806680797087464151, −16.81836087697764046457615963010, −16.32735523484281945927394087164, −15.28703730906165985756486928094, −14.795146477630237600654892418434, −14.389840917060212599459699810430, −13.5966233772569151440419737480, −13.24365509851268829027880599588, −12.10542664279298902538834617786, −11.94061815099920202196185315333, −11.15122304315494489645864223311, −9.85228759135205656944913765048, −9.7017084283774767363997201497, −8.85881685524175518523429622443, −7.52961811052002697757200990825, −7.00742866153250539543302149839, −6.63235478802382518890471023202, −5.85285699554288839544699311778, −4.92953308281630310921368601272, −3.71339252977400828001515065166, −3.37711583562243226724632226802, −2.65848252986633221197076926032, −2.00188643200389431047479382255, −1.090130926672283655614314429208, 1.021737945943596847176234315557, 1.91281655958789201550995791338, 3.07894715595173626711659943680, 3.47833601858915828063216826252, 4.14932292261458613272575007874, 5.02831838299000571318271468398, 5.523571192348255731310418386298, 6.46160475961189402844003119730, 7.34529938312374321495991864763, 7.96429773674103933707074757613, 8.7576233129896797149860811731, 9.5769823855737756050916728341, 10.04904983010480204528839591670, 10.98003388464121293386957969626, 11.85014554189223921665432730408, 12.76813767155492435185869788107, 12.99086068167411697255243031720, 13.75469586105754282038002560840, 14.40437364793533781553113156309, 15.04882156164115706089965013556, 15.69984220044833476792594393903, 16.44179388240502876517136392258, 16.80726414227090203763960956223, 17.36001644664651790313181142246, 18.78419800399916402402282323859

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