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Properties

Label 1-4029-4029.410-r0-0-0
Degree $1$
Conductor $4029$
Sign $0.687 - 0.726i$
Analytic cond. $18.7105$
Root an. cond. $18.7105$
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  + (0.992 − 0.120i)2-s + (0.970 − 0.239i)4-s + (0.954 − 0.297i)5-s + (−0.911 + 0.410i)7-s + (0.935 − 0.354i)8-s + (0.911 − 0.410i)10-s + (−0.954 − 0.297i)11-s + (0.970 − 0.239i)13-s + (−0.855 + 0.517i)14-s + (0.885 − 0.464i)16-s + (0.822 + 0.568i)19-s + (0.855 − 0.517i)20-s + (−0.983 − 0.180i)22-s + (−0.707 − 0.707i)23-s + (0.822 − 0.568i)25-s + (0.935 − 0.354i)26-s + ⋯
L(s)  = 1  + (0.992 − 0.120i)2-s + (0.970 − 0.239i)4-s + (0.954 − 0.297i)5-s + (−0.911 + 0.410i)7-s + (0.935 − 0.354i)8-s + (0.911 − 0.410i)10-s + (−0.954 − 0.297i)11-s + (0.970 − 0.239i)13-s + (−0.855 + 0.517i)14-s + (0.885 − 0.464i)16-s + (0.822 + 0.568i)19-s + (0.855 − 0.517i)20-s + (−0.983 − 0.180i)22-s + (−0.707 − 0.707i)23-s + (0.822 − 0.568i)25-s + (0.935 − 0.354i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.687 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.687 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4029\)    =    \(3 \cdot 17 \cdot 79\)
Sign: $0.687 - 0.726i$
Analytic conductor: \(18.7105\)
Root analytic conductor: \(18.7105\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4029} (410, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4029,\ (0:\ ),\ 0.687 - 0.726i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.716039945 - 1.599172894i\)
\(L(\frac12)\) \(\approx\) \(3.716039945 - 1.599172894i\)
\(L(1)\) \(\approx\) \(2.160774793 - 0.4126373514i\)
\(L(1)\) \(\approx\) \(2.160774793 - 0.4126373514i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
79 \( 1 \)
good2 \( 1 + (0.992 - 0.120i)T \)
5 \( 1 + (0.954 - 0.297i)T \)
7 \( 1 + (-0.911 + 0.410i)T \)
11 \( 1 + (-0.954 - 0.297i)T \)
13 \( 1 + (0.970 - 0.239i)T \)
19 \( 1 + (0.822 + 0.568i)T \)
23 \( 1 + (-0.707 - 0.707i)T \)
29 \( 1 + (0.0603 - 0.998i)T \)
31 \( 1 + (0.787 + 0.616i)T \)
37 \( 1 + (0.180 + 0.983i)T \)
41 \( 1 + (-0.297 - 0.954i)T \)
43 \( 1 + (-0.464 + 0.885i)T \)
47 \( 1 + (-0.568 - 0.822i)T \)
53 \( 1 + (0.935 - 0.354i)T \)
59 \( 1 + (-0.239 + 0.970i)T \)
61 \( 1 + (0.983 - 0.180i)T \)
67 \( 1 + (0.120 - 0.992i)T \)
71 \( 1 + (-0.911 - 0.410i)T \)
73 \( 1 + (0.855 - 0.517i)T \)
83 \( 1 + (0.239 + 0.970i)T \)
89 \( 1 + (-0.354 + 0.935i)T \)
97 \( 1 + (0.180 + 0.983i)T \)
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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

−18.4813714606846735557302102147, −17.846447410422831770381089554368, −17.10690839559276123318781412607, −16.19832710641333343132248809325, −15.913315502678750918922849792665, −15.16627302624293478455690875660, −14.20404285137111274569647336652, −13.751757185317530789040406239993, −13.10718707482757600929418943116, −12.83707372509077850897163741773, −11.72463585835987340838999712612, −11.0610717493924444522345528865, −10.269866172205046871892367732565, −9.84111687710221932411888150658, −8.88787653696885039837142878129, −7.82332897052948749280057361415, −7.08993603566096835885821460162, −6.509777056718931636430503388019, −5.77953626377503142618424662903, −5.27178245468156739521782920354, −4.30149038940084714310878077631, −3.42675862775023740196269010621, −2.86332055853911819783088670152, −2.063810835729299544101422279311, −1.114115118815402584368735855502, 0.81092061247831319746953276089, 1.83169421699443836627786280355, 2.64564011974937184333568792306, 3.20041554095975408493015126616, 4.066845550184126452849902252599, 5.11946691034490739553586815514, 5.595509879615049947393894007748, 6.26521027385812841965953032442, 6.717407908763783798532036801860, 7.96005912916793634102239324366, 8.53879043329617132301472152605, 9.68821136833482347043190528675, 10.14901493085251728992689386663, 10.733045277975560515235476859787, 11.80002376875376665656263450417, 12.32212360897788000498036466795, 13.11606488997386901138566473678, 13.565175802501433066409094543494, 13.935274436697005804506790812534, 15.03702643359605863469704504314, 15.60518967262877910377913624118, 16.374123016400090720755185629611, 16.56707267649368556322892297569, 17.83933548164990682305190299622, 18.3992408391779446897254124746

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