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Properties

Label 1-4029-4029.1172-r0-0-0
Degree $1$
Conductor $4029$
Sign $0.307 + 0.951i$
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  + (0.0402 + 0.999i)2-s + (−0.996 + 0.0804i)4-s + (0.987 + 0.160i)5-s + (0.799 + 0.600i)7-s + (−0.120 − 0.992i)8-s + (−0.120 + 0.992i)10-s + (−0.632 − 0.774i)11-s + (0.428 − 0.903i)13-s + (−0.568 + 0.822i)14-s + (0.987 − 0.160i)16-s + (0.948 − 0.316i)19-s + (−0.996 − 0.0804i)20-s + (0.748 − 0.663i)22-s + (−0.5 + 0.866i)23-s + (0.948 + 0.316i)25-s + (0.919 + 0.391i)26-s + ⋯
L(s)  = 1  + (0.0402 + 0.999i)2-s + (−0.996 + 0.0804i)4-s + (0.987 + 0.160i)5-s + (0.799 + 0.600i)7-s + (−0.120 − 0.992i)8-s + (−0.120 + 0.992i)10-s + (−0.632 − 0.774i)11-s + (0.428 − 0.903i)13-s + (−0.568 + 0.822i)14-s + (0.987 − 0.160i)16-s + (0.948 − 0.316i)19-s + (−0.996 − 0.0804i)20-s + (0.748 − 0.663i)22-s + (−0.5 + 0.866i)23-s + (0.948 + 0.316i)25-s + (0.919 + 0.391i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.307 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.307 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4029\)    =    \(3 \cdot 17 \cdot 79\)
Sign: $0.307 + 0.951i$
Analytic conductor: \(18.7105\)
Root analytic conductor: \(18.7105\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4029} (1172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4029,\ (0:\ ),\ 0.307 + 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.821530105 + 1.325235885i\)
\(L(\frac12)\) \(\approx\) \(1.821530105 + 1.325235885i\)
\(L(1)\) \(\approx\) \(1.164392948 + 0.6185974496i\)
\(L(1)\) \(\approx\) \(1.164392948 + 0.6185974496i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
79 \( 1 \)
good2 \( 1 + (0.0402 + 0.999i)T \)
5 \( 1 + (0.987 + 0.160i)T \)
7 \( 1 + (0.799 + 0.600i)T \)
11 \( 1 + (-0.632 - 0.774i)T \)
13 \( 1 + (0.428 - 0.903i)T \)
19 \( 1 + (0.948 - 0.316i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.278 + 0.960i)T \)
31 \( 1 + (0.845 - 0.534i)T \)
37 \( 1 + (-0.200 + 0.979i)T \)
41 \( 1 + (0.354 + 0.935i)T \)
43 \( 1 + (0.632 - 0.774i)T \)
47 \( 1 + (-0.200 - 0.979i)T \)
53 \( 1 + (-0.919 - 0.391i)T \)
59 \( 1 + (-0.996 - 0.0804i)T \)
61 \( 1 + (-0.748 - 0.663i)T \)
67 \( 1 + (0.885 - 0.464i)T \)
71 \( 1 + (-0.120 - 0.992i)T \)
73 \( 1 + (-0.428 - 0.903i)T \)
83 \( 1 + (0.996 - 0.0804i)T \)
89 \( 1 + (-0.120 + 0.992i)T \)
97 \( 1 + (0.748 + 0.663i)T \)
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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

−18.361213635531933045936567675847, −17.65338755708697067901268553827, −17.38217529997386589240171984363, −16.47130996550836336048608212684, −15.63484035765243529399803134567, −14.37174936581822901259859860505, −14.20751568838715183147006582250, −13.5861740726513493147784304373, −12.79220041224918298548962171129, −12.20808465414965593982248657920, −11.374204764973154029839244739012, −10.72297298120733195916035417543, −10.10260967554990153410312163981, −9.539314581763366864480327842919, −8.81198663768936203846669905396, −7.99592705407926017284564906683, −7.263901796477230308819189669473, −6.13826311590654614769417373858, −5.451209406262073026212605818690, −4.50853154397072850038939117190, −4.296170002239590786440849995185, −3.03939905318982425947537718665, −2.19885830461697186452970889691, −1.65797826541264008005737522462, −0.855274866747927496440791100805, 0.83135464389117645383382726265, 1.74843913149557642422586333185, 2.91648893577215225716975062596, 3.48522544685328355953109360102, 4.97698869357326400668185826344, 5.14197251389779148408030158378, 5.96259047608007561677827714662, 6.401569952781631147338764876481, 7.63833807646183768022658317945, 7.9668675840432731826047285027, 8.82680869805353098590851134497, 9.394936099626522287127788596395, 10.21834383714600172345243860298, 10.899476419016598274476457505561, 11.79657348425124732377354855157, 12.72191280284868247202455051758, 13.48506540027216684539923847588, 13.801412940540717056226282410458, 14.53839390043325223648048947312, 15.35085364522072211777623167209, 15.72211479841055012127867792631, 16.58720785186210575843469072014, 17.33017310735185515817470334477, 17.8978471799172552288539055702, 18.349590745444651317549202066934

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