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Properties

Label 1-4029-4029.143-r0-0-0
Degree $1$
Conductor $4029$
Sign $0.414 + 0.909i$
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  + (−0.616 + 0.787i)2-s + (−0.239 − 0.970i)4-s + (0.995 + 0.0904i)5-s + (0.685 − 0.728i)7-s + (0.911 + 0.410i)8-s + (−0.685 + 0.728i)10-s + (0.0904 + 0.995i)11-s + (0.239 + 0.970i)13-s + (0.150 + 0.988i)14-s + (−0.885 + 0.464i)16-s + (−0.983 + 0.180i)19-s + (−0.150 − 0.988i)20-s + (−0.839 − 0.542i)22-s + (0.382 − 0.923i)23-s + (0.983 + 0.180i)25-s + (−0.911 − 0.410i)26-s + ⋯
L(s)  = 1  + (−0.616 + 0.787i)2-s + (−0.239 − 0.970i)4-s + (0.995 + 0.0904i)5-s + (0.685 − 0.728i)7-s + (0.911 + 0.410i)8-s + (−0.685 + 0.728i)10-s + (0.0904 + 0.995i)11-s + (0.239 + 0.970i)13-s + (0.150 + 0.988i)14-s + (−0.885 + 0.464i)16-s + (−0.983 + 0.180i)19-s + (−0.150 − 0.988i)20-s + (−0.839 − 0.542i)22-s + (0.382 − 0.923i)23-s + (0.983 + 0.180i)25-s + (−0.911 − 0.410i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.414 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.414 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4029\)    =    \(3 \cdot 17 \cdot 79\)
Sign: $0.414 + 0.909i$
Analytic conductor: \(18.7105\)
Root analytic conductor: \(18.7105\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4029} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4029,\ (0:\ ),\ 0.414 + 0.909i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.412051231 + 0.9079477012i\)
\(L(\frac12)\) \(\approx\) \(1.412051231 + 0.9079477012i\)
\(L(1)\) \(\approx\) \(0.9680716187 + 0.3512693096i\)
\(L(1)\) \(\approx\) \(0.9680716187 + 0.3512693096i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
79 \( 1 \)
good2 \( 1 + (-0.616 + 0.787i)T \)
5 \( 1 + (0.995 + 0.0904i)T \)
7 \( 1 + (0.685 - 0.728i)T \)
11 \( 1 + (0.0904 + 0.995i)T \)
13 \( 1 + (0.239 + 0.970i)T \)
19 \( 1 + (-0.983 + 0.180i)T \)
23 \( 1 + (0.382 - 0.923i)T \)
29 \( 1 + (0.437 - 0.899i)T \)
31 \( 1 + (-0.871 + 0.491i)T \)
37 \( 1 + (-0.977 - 0.209i)T \)
41 \( 1 + (0.640 + 0.768i)T \)
43 \( 1 + (-0.954 + 0.297i)T \)
47 \( 1 + (-0.822 + 0.568i)T \)
53 \( 1 + (0.410 - 0.911i)T \)
59 \( 1 + (0.855 - 0.517i)T \)
61 \( 1 + (0.839 - 0.542i)T \)
67 \( 1 + (-0.120 + 0.992i)T \)
71 \( 1 + (0.728 - 0.685i)T \)
73 \( 1 + (0.988 - 0.150i)T \)
83 \( 1 + (0.855 + 0.517i)T \)
89 \( 1 + (0.935 + 0.354i)T \)
97 \( 1 + (0.209 - 0.977i)T \)
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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

−18.31212373265598605764944650311, −17.788363549971999548559115501055, −17.21117973253664859615047542607, −16.60156232513108294995060808509, −15.72979099760842376402819336276, −14.91794907434701038980463795457, −14.093404687689523456033510473720, −13.36763700383526208985183361110, −12.867612631349426457432240352196, −12.11353319382241398623991324060, −11.2950126763668554635413864725, −10.734773299734977311031507967684, −10.19373572264891423372823531592, −9.21408957164806967254480733322, −8.72824685753687373233013136632, −8.26992374372849677667251547132, −7.30195693364043711423959997536, −6.34368820249758900167089085544, −5.45954951371033758158350434822, −5.02022278112391635230588564569, −3.75728958090223192737352841772, −3.060514007813593898032056403091, −2.237537259105490919650906659038, −1.61093532016648606692257025688, −0.70967798125184202785756958537, 0.8831337361764027530894344784, 1.88995609971732592474377584621, 2.125957691500568355195925717554, 3.80147107603833430327485263275, 4.73013235514031477001725009346, 5.02992966457009748990666056550, 6.2532889109186234546770814203, 6.65881223356754728070687927703, 7.26539349478892650926038458179, 8.228442243898145781367632893299, 8.77907159236016542620312482083, 9.64206842246796866154816546667, 10.11310953204729697169137452919, 10.80373510460039358836479124823, 11.451197232731722355136499737952, 12.65216552620226426566812227808, 13.31456974013248313519374303168, 14.1170911630405487890630081700, 14.55415118627861893305200596982, 15.01035805800252736662010459576, 16.12352920590465573508067784959, 16.72067060432725104975105547807, 17.24089616249934573217901845329, 17.82281068136683264631482526680, 18.28309141548457710718329376335

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