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Properties

Label 1-4031-4031.75-r0-0-0
Degree $1$
Conductor $4031$
Sign $-0.992 + 0.121i$
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  + (−0.269 + 0.962i)2-s + (0.979 + 0.203i)3-s + (−0.854 − 0.519i)4-s + (0.0682 + 0.997i)5-s + (−0.460 + 0.887i)6-s + (0.460 + 0.887i)7-s + (0.730 − 0.682i)8-s + (0.917 + 0.398i)9-s + (−0.979 − 0.203i)10-s + (0.942 + 0.334i)11-s + (−0.730 − 0.682i)12-s + (−0.203 + 0.979i)13-s + (−0.979 + 0.203i)14-s + (−0.136 + 0.990i)15-s + (0.460 + 0.887i)16-s + (0.816 − 0.576i)17-s + ⋯
L(s)  = 1  + (−0.269 + 0.962i)2-s + (0.979 + 0.203i)3-s + (−0.854 − 0.519i)4-s + (0.0682 + 0.997i)5-s + (−0.460 + 0.887i)6-s + (0.460 + 0.887i)7-s + (0.730 − 0.682i)8-s + (0.917 + 0.398i)9-s + (−0.979 − 0.203i)10-s + (0.942 + 0.334i)11-s + (−0.730 − 0.682i)12-s + (−0.203 + 0.979i)13-s + (−0.979 + 0.203i)14-s + (−0.136 + 0.990i)15-s + (0.460 + 0.887i)16-s + (0.816 − 0.576i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.121i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.121i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4031\)    =    \(29 \cdot 139\)
Sign: $-0.992 + 0.121i$
Analytic conductor: \(18.7198\)
Root analytic conductor: \(18.7198\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4031} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4031,\ (0:\ ),\ -0.992 + 0.121i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1493013238 + 2.457230154i\)
\(L(\frac12)\) \(\approx\) \(0.1493013238 + 2.457230154i\)
\(L(1)\) \(\approx\) \(0.9241501508 + 1.079688981i\)
\(L(1)\) \(\approx\) \(0.9241501508 + 1.079688981i\)

Euler product

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

−18.29438338219054582952031320798, −17.42046471605173198200087600985, −17.05301980011431699125471933897, −16.31829976247597118340067694219, −15.29995290626568533239466129227, −14.435970142642873777437115198846, −13.84671869616859377893108087503, −13.36947232621056750785264097878, −12.68438110270685638723860714482, −11.98853034261338131698244653683, −11.43202524224098350888407122902, −10.22722170583120180409972141487, −9.88248748076448008489198132322, −9.2419245637094401817053399546, −8.280614001383599024214403585391, −8.018597444420903169228228061, −7.40743325663164095449851000841, −6.07228045832251114688966295985, −5.06779303649659853026001589917, −4.34781249389768489076280161125, −3.53632084883350837633256363726, −3.23756485479061261029254863785, −1.78935466669371855614250552238, −1.420706253399818671884977679835, −0.697975351728572043310673452108, 1.33280576861237441705212565220, 2.11339389234217640831145970925, 2.9984894155779110692993185433, 3.84964606790041056267752154633, 4.66218268408298454010237923633, 5.36738355088457651014512444702, 6.47298821126612318183777382158, 6.895597057990442392788267190425, 7.61918400472059905316741191397, 8.326617245714397354831019787882, 9.09320333505638228595780782552, 9.5606965637325186814245463386, 10.13454489734173418089155984308, 11.17050649564438841183041463066, 11.94851557401330419544502727433, 12.787220914724188731208234821881, 14.00906307791252660730912403711, 14.23898975340406748627392707469, 14.53063877431276444411396703426, 15.391129036839644385327044399001, 15.88234956535513205727309133858, 16.59491855347682146776275983216, 17.57033490866795465281532991542, 18.20846699243949763544715391591, 18.6557946973294895561793224424

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