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Properties

Label 1-4029-4029.926-r0-0-0
Degree $1$
Conductor $4029$
Sign $-0.279 - 0.960i$
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  + (0.935 − 0.354i)2-s + (0.748 − 0.663i)4-s + (0.616 − 0.787i)5-s + (−0.297 + 0.954i)7-s + (0.464 − 0.885i)8-s + (0.297 − 0.954i)10-s + (−0.616 − 0.787i)11-s + (0.748 − 0.663i)13-s + (0.0603 + 0.998i)14-s + (0.120 − 0.992i)16-s + (−0.239 + 0.970i)19-s + (−0.0603 − 0.998i)20-s + (−0.855 − 0.517i)22-s + (0.707 − 0.707i)23-s + (−0.239 − 0.970i)25-s + (0.464 − 0.885i)26-s + ⋯
L(s)  = 1  + (0.935 − 0.354i)2-s + (0.748 − 0.663i)4-s + (0.616 − 0.787i)5-s + (−0.297 + 0.954i)7-s + (0.464 − 0.885i)8-s + (0.297 − 0.954i)10-s + (−0.616 − 0.787i)11-s + (0.748 − 0.663i)13-s + (0.0603 + 0.998i)14-s + (0.120 − 0.992i)16-s + (−0.239 + 0.970i)19-s + (−0.0603 − 0.998i)20-s + (−0.855 − 0.517i)22-s + (0.707 − 0.707i)23-s + (−0.239 − 0.970i)25-s + (0.464 − 0.885i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.279 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.279 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4029\)    =    \(3 \cdot 17 \cdot 79\)
Sign: $-0.279 - 0.960i$
Analytic conductor: \(18.7105\)
Root analytic conductor: \(18.7105\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4029} (926, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4029,\ (0:\ ),\ -0.279 - 0.960i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.135365322 - 2.846304977i\)
\(L(\frac12)\) \(\approx\) \(2.135365322 - 2.846304977i\)
\(L(1)\) \(\approx\) \(1.807545907 - 0.8971039374i\)
\(L(1)\) \(\approx\) \(1.807545907 - 0.8971039374i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
79 \( 1 \)
good2 \( 1 + (0.935 - 0.354i)T \)
5 \( 1 + (0.616 - 0.787i)T \)
7 \( 1 + (-0.297 + 0.954i)T \)
11 \( 1 + (-0.616 - 0.787i)T \)
13 \( 1 + (0.748 - 0.663i)T \)
19 \( 1 + (-0.239 + 0.970i)T \)
23 \( 1 + (0.707 - 0.707i)T \)
29 \( 1 + (-0.180 + 0.983i)T \)
31 \( 1 + (-0.410 + 0.911i)T \)
37 \( 1 + (-0.517 - 0.855i)T \)
41 \( 1 + (0.787 + 0.616i)T \)
43 \( 1 + (0.992 - 0.120i)T \)
47 \( 1 + (0.970 - 0.239i)T \)
53 \( 1 + (0.464 - 0.885i)T \)
59 \( 1 + (0.663 - 0.748i)T \)
61 \( 1 + (0.855 - 0.517i)T \)
67 \( 1 + (-0.354 + 0.935i)T \)
71 \( 1 + (-0.297 - 0.954i)T \)
73 \( 1 + (-0.0603 - 0.998i)T \)
83 \( 1 + (-0.663 - 0.748i)T \)
89 \( 1 + (0.885 - 0.464i)T \)
97 \( 1 + (-0.517 - 0.855i)T \)
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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

−18.733829528806939614501073779871, −17.60627746949011853376903346789, −17.383152756969861633763127428480, −16.605527517600706627706692071124, −15.700935253477064402979293524957, −15.27731663368223491540954558772, −14.52197030208728770573324701475, −13.73438036222353209986212018386, −13.40498406100717853577405389057, −12.87759821581802506782564598504, −11.81980201317299794439873884702, −11.03672919651519758712016552216, −10.67258957419492752325631830535, −9.76802994858904024413575857818, −9.01748067461233275098308780317, −7.8231460418632508473095803488, −7.18112701356860424242703274656, −6.81521402954984727958891819881, −5.97040545429187014818964646127, −5.32485103840548012565484818404, −4.25164975147585344742835031464, −3.90190255361520398414142672312, −2.78821691459418921728531449074, −2.321629691822018583597867057865, −1.24544098171732447854081980345, 0.6856442825970453149989230016, 1.61550685595503577023652730840, 2.426142857594505138783192272531, 3.157859800658676725020959377476, 3.88946148189440586490395000570, 4.99863479397230456862037744527, 5.51219896832578863432388251016, 5.92694064343605787542459699834, 6.689953242250396219729365721396, 7.84960651973252375642679240242, 8.7148909318513725825298156724, 9.14673325265453528380559319297, 10.29559667635942368737367612209, 10.63768743553239670169029113173, 11.529829912582178227112689840877, 12.43350719723926610250180921891, 12.802746602355622248009173544749, 13.23456155127235467194640950982, 14.19771562708017387785891852826, 14.637847288332789103769289707230, 15.655298894750716638637926201039, 16.12901527797729570246394221671, 16.53751486227259282898744051370, 17.71415197299290456988235691526, 18.39379754410434217174614264015

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