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Properties

Label 1-4031-4031.8-r0-0-0
Degree $1$
Conductor $4031$
Sign $0.968 - 0.247i$
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  + (0.689 + 0.724i)2-s + (−0.896 + 0.442i)3-s + (−0.0487 + 0.998i)4-s + (0.145 + 0.989i)5-s + (−0.938 − 0.344i)6-s + (−0.696 + 0.717i)7-s + (−0.756 + 0.653i)8-s + (0.608 − 0.793i)9-s + (−0.615 + 0.787i)10-s + (−0.485 + 0.874i)11-s + (−0.398 − 0.917i)12-s + (−0.425 + 0.905i)13-s + (−0.999 − 0.00975i)14-s + (−0.568 − 0.822i)15-s + (−0.995 − 0.0974i)16-s + (−0.887 − 0.460i)17-s + ⋯
L(s)  = 1  + (0.689 + 0.724i)2-s + (−0.896 + 0.442i)3-s + (−0.0487 + 0.998i)4-s + (0.145 + 0.989i)5-s + (−0.938 − 0.344i)6-s + (−0.696 + 0.717i)7-s + (−0.756 + 0.653i)8-s + (0.608 − 0.793i)9-s + (−0.615 + 0.787i)10-s + (−0.485 + 0.874i)11-s + (−0.398 − 0.917i)12-s + (−0.425 + 0.905i)13-s + (−0.999 − 0.00975i)14-s + (−0.568 − 0.822i)15-s + (−0.995 − 0.0974i)16-s + (−0.887 − 0.460i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4031\)    =    \(29 \cdot 139\)
Sign: $0.968 - 0.247i$
Analytic conductor: \(18.7198\)
Root analytic conductor: \(18.7198\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4031} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4031,\ (0:\ ),\ 0.968 - 0.247i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.5881783756 + 0.07401242507i\)
\(L(\frac12)\) \(\approx\) \(-0.5881783756 + 0.07401242507i\)
\(L(1)\) \(\approx\) \(0.3369688614 + 0.7406308797i\)
\(L(1)\) \(\approx\) \(0.3369688614 + 0.7406308797i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 \)
139 \( 1 \)
good2 \( 1 + (0.689 + 0.724i)T \)
3 \( 1 + (-0.896 + 0.442i)T \)
5 \( 1 + (0.145 + 0.989i)T \)
7 \( 1 + (-0.696 + 0.717i)T \)
11 \( 1 + (-0.485 + 0.874i)T \)
13 \( 1 + (-0.425 + 0.905i)T \)
17 \( 1 + (-0.887 - 0.460i)T \)
19 \( 1 + (-0.250 + 0.967i)T \)
23 \( 1 + (-0.126 - 0.991i)T \)
31 \( 1 + (-0.451 + 0.892i)T \)
37 \( 1 + (0.928 + 0.371i)T \)
41 \( 1 + (-0.942 + 0.334i)T \)
43 \( 1 + (0.781 - 0.623i)T \)
47 \( 1 + (-0.362 + 0.932i)T \)
53 \( 1 + (0.724 - 0.689i)T \)
59 \( 1 + (-0.962 + 0.269i)T \)
61 \( 1 + (-0.536 + 0.844i)T \)
67 \( 1 + (-0.874 + 0.485i)T \)
71 \( 1 + (0.297 - 0.954i)T \)
73 \( 1 + (0.965 - 0.260i)T \)
79 \( 1 + (0.703 + 0.710i)T \)
83 \( 1 + (-0.668 + 0.744i)T \)
89 \( 1 + (-0.675 + 0.737i)T \)
97 \( 1 + (0.974 - 0.222i)T \)
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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

−17.818774638458632689444937901359, −17.11648074066062303691034751075, −16.56639388892017849174241681814, −15.6506678443435135612994215430, −15.357460214185623194991269430571, −13.91972114657315367604645796894, −13.369586706934278561285998721394, −12.99929171706525036163243227651, −12.58189174587282036853441664163, −11.62119062343870953497913339003, −11.054093345694235966313797116653, −10.450978428320046956500792930465, −9.73067862526285894516401169826, −8.984236329608483990250024358669, −7.92909103849734431208977122607, −7.14321648177039697339663414623, −6.138093417189859626228069323234, −5.78569721784891652650913401253, −4.97091019522071195897313989282, −4.376597513408242334139672470769, −3.52111555488848455418810872189, −2.54297016461214566050730382208, −1.66227894491939554525563181159, −0.66020502118404114479355706951, −0.21132373027172953053202578808, 2.01200846815930169107796974680, 2.66603099010950269827785968723, 3.57502337809848120074960211185, 4.39327769549518256767766615399, 4.97918615063998687739262874326, 5.872185211772223068109013643463, 6.45164230209951865382254201108, 6.87759669437195930063140591497, 7.59378756318400236583658241630, 8.76506829932348164255480378256, 9.504037575168533760443750437904, 10.16795850267688585538463360641, 10.96289392904702264865195971170, 11.77043065513794197230899774288, 12.31149397717062927573380468716, 12.852860968499464571095387782642, 13.797125165990512654387757399571, 14.57417649279231122726388356126, 15.17319682341987090669127521263, 15.574171916338735754665448871280, 16.456180861208103903773848918333, 16.74544492276672159211187573797, 17.86456063930773045265826678640, 18.15260384525323607930823815123, 18.762515305325847692307424738400

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