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Properties

Label 1-4029-4029.848-r0-0-0
Degree $1$
Conductor $4029$
Sign $-0.995 + 0.0949i$
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 + (−0.297 + 0.954i)5-s + (−0.410 + 0.911i)7-s + (−0.935 − 0.354i)8-s + (0.410 − 0.911i)10-s + (0.297 + 0.954i)11-s + (0.970 + 0.239i)13-s + (0.517 − 0.855i)14-s + (0.885 + 0.464i)16-s + (−0.822 + 0.568i)19-s + (−0.517 + 0.855i)20-s + (−0.180 − 0.983i)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 + (−0.297 + 0.954i)5-s + (−0.410 + 0.911i)7-s + (−0.935 − 0.354i)8-s + (0.410 − 0.911i)10-s + (0.297 + 0.954i)11-s + (0.970 + 0.239i)13-s + (0.517 − 0.855i)14-s + (0.885 + 0.464i)16-s + (−0.822 + 0.568i)19-s + (−0.517 + 0.855i)20-s + (−0.180 − 0.983i)22-s + (0.707 + 0.707i)23-s + (−0.822 − 0.568i)25-s + (−0.935 − 0.354i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4029\)    =    \(3 \cdot 17 \cdot 79\)
Sign: $-0.995 + 0.0949i$
Analytic conductor: \(18.7105\)
Root analytic conductor: \(18.7105\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4029} (848, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4029,\ (0:\ ),\ -0.995 + 0.0949i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04069904503 + 0.8551678599i\)
\(L(\frac12)\) \(\approx\) \(0.04069904503 + 0.8551678599i\)
\(L(1)\) \(\approx\) \(0.5790249278 + 0.3268733969i\)
\(L(1)\) \(\approx\) \(0.5790249278 + 0.3268733969i\)

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 \( 1 + (-0.297 + 0.954i)T \)
7 \( 1 + (-0.410 + 0.911i)T \)
11 \( 1 + (0.297 + 0.954i)T \)
13 \( 1 + (0.970 + 0.239i)T \)
19 \( 1 + (-0.822 + 0.568i)T \)
23 \( 1 + (0.707 + 0.707i)T \)
29 \( 1 + (-0.998 + 0.0603i)T \)
31 \( 1 + (0.616 + 0.787i)T \)
37 \( 1 + (0.983 + 0.180i)T \)
41 \( 1 + (0.954 + 0.297i)T \)
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 \( 1 + (0.180 - 0.983i)T \)
67 \( 1 + (0.120 + 0.992i)T \)
71 \( 1 + (-0.410 - 0.911i)T \)
73 \( 1 + (-0.517 + 0.855i)T \)
83 \( 1 + (-0.239 + 0.970i)T \)
89 \( 1 + (-0.354 - 0.935i)T \)
97 \( 1 + (0.983 + 0.180i)T \)
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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

−18.09912657679636298428586560034, −17.22801667923078140232400788466, −16.82146455659190859529309913316, −16.32750421854823062871833683893, −15.709127509963701397945086636, −14.99211917890686749722224156028, −14.04163181120572594594893780734, −13.16341059844866340519321369561, −12.80642482401000249948178925318, −11.70003420629452362346083030841, −11.06314979648503517076403161775, −10.65310153042809696174266315457, −9.64728973103103876948281739267, −9.02550548344732736902587395913, −8.46625231928207611708445797396, −7.840468677381399409995928655446, −7.04345228487783841717230987358, −6.221156479114044510688541685094, −5.70262047947268124527312053517, −4.50860652660136982305039394408, −3.786225200080227012348651497309, −2.9644114876005182088965318657, −1.83200584121590744470280700995, −0.83332580454504742178890412366, −0.45192124732541626157992780144, 1.27486057570611732103344361870, 2.05766165419251538075210313736, 2.83801180969940116654531318562, 3.49476907405160284257284728634, 4.409172303832970280464987311363, 5.80144526656188187030420878122, 6.30661918726615414232622224811, 6.93061830381307760417197897515, 7.71417947327522007405727897410, 8.377402814440155472541727179687, 9.2471357136391965142276155022, 9.68259117572916819839775966533, 10.49811162063263815310228517187, 11.26260573030869031781596144895, 11.61042500631485691275680602589, 12.57046300360403500518886983270, 13.03190832300471511598001544929, 14.40634797993208287765524723922, 14.857184887565857786573695163036, 15.545461269483580384720576169253, 16.017728018535784738152603529029, 16.83347631596763287194560834050, 17.7614736379728284184586586172, 18.06644643777167691932046073223, 18.91384428931415594176741875050

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