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Properties

Label 1-4029-4029.308-r0-0-0
Degree $1$
Conductor $4029$
Sign $-0.937 - 0.347i$
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  + (−0.663 + 0.748i)2-s + (−0.120 − 0.992i)4-s + (−0.855 + 0.517i)5-s + (−0.180 + 0.983i)7-s + (0.822 + 0.568i)8-s + (0.180 − 0.983i)10-s + (0.855 + 0.517i)11-s + (−0.120 − 0.992i)13-s + (−0.616 − 0.787i)14-s + (−0.970 + 0.239i)16-s + (0.464 + 0.885i)19-s + (0.616 + 0.787i)20-s + (−0.954 + 0.297i)22-s + (−0.707 − 0.707i)23-s + (0.464 − 0.885i)25-s + (0.822 + 0.568i)26-s + ⋯
L(s)  = 1  + (−0.663 + 0.748i)2-s + (−0.120 − 0.992i)4-s + (−0.855 + 0.517i)5-s + (−0.180 + 0.983i)7-s + (0.822 + 0.568i)8-s + (0.180 − 0.983i)10-s + (0.855 + 0.517i)11-s + (−0.120 − 0.992i)13-s + (−0.616 − 0.787i)14-s + (−0.970 + 0.239i)16-s + (0.464 + 0.885i)19-s + (0.616 + 0.787i)20-s + (−0.954 + 0.297i)22-s + (−0.707 − 0.707i)23-s + (0.464 − 0.885i)25-s + (0.822 + 0.568i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.937 - 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.937 - 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4029\)    =    \(3 \cdot 17 \cdot 79\)
Sign: $-0.937 - 0.347i$
Analytic conductor: \(18.7105\)
Root analytic conductor: \(18.7105\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4029} (308, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4029,\ (0:\ ),\ -0.937 - 0.347i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.09568802131 + 0.5335116905i\)
\(L(\frac12)\) \(\approx\) \(-0.09568802131 + 0.5335116905i\)
\(L(1)\) \(\approx\) \(0.5187242945 + 0.3403239487i\)
\(L(1)\) \(\approx\) \(0.5187242945 + 0.3403239487i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
79 \( 1 \)
good2 \( 1 + (-0.663 + 0.748i)T \)
5 \( 1 + (-0.855 + 0.517i)T \)
7 \( 1 + (-0.180 + 0.983i)T \)
11 \( 1 + (0.855 + 0.517i)T \)
13 \( 1 + (-0.120 - 0.992i)T \)
19 \( 1 + (0.464 + 0.885i)T \)
23 \( 1 + (-0.707 - 0.707i)T \)
29 \( 1 + (0.911 + 0.410i)T \)
31 \( 1 + (-0.998 + 0.0603i)T \)
37 \( 1 + (-0.297 + 0.954i)T \)
41 \( 1 + (0.517 + 0.855i)T \)
43 \( 1 + (0.239 - 0.970i)T \)
47 \( 1 + (-0.885 - 0.464i)T \)
53 \( 1 + (0.822 + 0.568i)T \)
59 \( 1 + (-0.992 - 0.120i)T \)
61 \( 1 + (0.954 + 0.297i)T \)
67 \( 1 + (-0.748 + 0.663i)T \)
71 \( 1 + (-0.180 - 0.983i)T \)
73 \( 1 + (0.616 + 0.787i)T \)
83 \( 1 + (0.992 - 0.120i)T \)
89 \( 1 + (0.568 + 0.822i)T \)
97 \( 1 + (-0.297 + 0.954i)T \)
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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

−18.038496881753699910515041442996, −17.457297980949033154135568686450, −16.74576015224365795114549851371, −16.22308141377728974689465677946, −15.80262094216550097448290235721, −14.51636113512782247877072425537, −13.84196786109616873031367380518, −13.22803477195951052889394618501, −12.39203737953624760678865974300, −11.74447204818432583329292887006, −11.255742269219425790640030936302, −10.6689794271607612999352843625, −9.56551680268252015892961906593, −9.24454800672686577047912580808, −8.46039643712211112811481557246, −7.64516360942313222126960099176, −7.15360390737455003667487560315, −6.37773174354535482388124712106, −5.01638261329961579364090352473, −4.14352390559622233979193177120, −3.84546179619218815309357908686, −3.02818562060668779403356842871, −1.83183427579739988109888167479, −1.06320210620211666157956950706, −0.247061648509227959871986334857, 1.04882134882422128939521443923, 2.07718846428700263365587838536, 2.99959184512485804802492695983, 3.900014280044697275015645119198, 4.82619749625622036885773971537, 5.61450587785921732282438196007, 6.36007074678938614827225997514, 6.94566605160673514749270205287, 7.7964673849122671809109672954, 8.30355613133811633223063435033, 8.98030731188441201725798728587, 9.85355021960630220073105668611, 10.368289259067875663231951564504, 11.19980523143912862432467109290, 12.10911242592315223437507034412, 12.372401962729566240949956137606, 13.61237655861826391079767696882, 14.54800631573788601226777051878, 14.85953897367114921708568066366, 15.43053572410749156278601387024, 16.194704193807973931807969522684, 16.58945570002096645479332970157, 17.74790844571588056973694567235, 18.061132802596593535263514712335, 18.75286441130918103907807646708

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