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Properties

Label 1-6043-6043.65-r0-0-0
Degree $1$
Conductor $6043$
Sign $0.241 + 0.970i$
Analytic cond. $28.0635$
Root an. cond. $28.0635$
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  + (−0.794 − 0.606i)2-s + (−0.987 − 0.158i)3-s + (0.263 + 0.964i)4-s + (−0.332 − 0.943i)5-s + (0.688 + 0.725i)6-s + (0.804 + 0.594i)7-s + (0.375 − 0.926i)8-s + (0.949 + 0.312i)9-s + (−0.308 + 0.951i)10-s + (0.945 + 0.324i)11-s + (−0.107 − 0.994i)12-s + (0.221 + 0.975i)13-s + (−0.278 − 0.960i)14-s + (0.178 + 0.983i)15-s + (−0.861 + 0.508i)16-s + (−0.824 − 0.566i)17-s + ⋯
L(s)  = 1  + (−0.794 − 0.606i)2-s + (−0.987 − 0.158i)3-s + (0.263 + 0.964i)4-s + (−0.332 − 0.943i)5-s + (0.688 + 0.725i)6-s + (0.804 + 0.594i)7-s + (0.375 − 0.926i)8-s + (0.949 + 0.312i)9-s + (−0.308 + 0.951i)10-s + (0.945 + 0.324i)11-s + (−0.107 − 0.994i)12-s + (0.221 + 0.975i)13-s + (−0.278 − 0.960i)14-s + (0.178 + 0.983i)15-s + (−0.861 + 0.508i)16-s + (−0.824 − 0.566i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6043 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.241 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6043 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.241 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6043\)
Sign: $0.241 + 0.970i$
Analytic conductor: \(28.0635\)
Root analytic conductor: \(28.0635\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6043} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6043,\ (0:\ ),\ 0.241 + 0.970i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4819019037 + 0.3767058696i\)
\(L(\frac12)\) \(\approx\) \(0.4819019037 + 0.3767058696i\)
\(L(1)\) \(\approx\) \(0.5717445175 - 0.08142847158i\)
\(L(1)\) \(\approx\) \(0.5717445175 - 0.08142847158i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad6043 \( 1 \)
good2 \( 1 + (-0.794 - 0.606i)T \)
3 \( 1 + (-0.987 - 0.158i)T \)
5 \( 1 + (-0.332 - 0.943i)T \)
7 \( 1 + (0.804 + 0.594i)T \)
11 \( 1 + (0.945 + 0.324i)T \)
13 \( 1 + (0.221 + 0.975i)T \)
17 \( 1 + (-0.824 - 0.566i)T \)
19 \( 1 + (0.432 + 0.901i)T \)
23 \( 1 + (0.977 + 0.210i)T \)
29 \( 1 + (-0.885 + 0.464i)T \)
31 \( 1 + (0.493 + 0.869i)T \)
37 \( 1 + (-0.999 + 0.0155i)T \)
41 \( 1 + (-0.384 - 0.923i)T \)
43 \( 1 + (0.983 + 0.179i)T \)
47 \( 1 + (-0.585 + 0.810i)T \)
53 \( 1 + (-0.355 - 0.934i)T \)
59 \( 1 + (-0.496 - 0.868i)T \)
61 \( 1 + (0.792 + 0.609i)T \)
67 \( 1 + (0.656 - 0.754i)T \)
71 \( 1 + (-0.992 - 0.121i)T \)
73 \( 1 + (-0.854 + 0.519i)T \)
79 \( 1 + (-0.605 + 0.795i)T \)
83 \( 1 + (0.999 + 0.0124i)T \)
89 \( 1 + (0.0171 + 0.999i)T \)
97 \( 1 + (-0.580 - 0.814i)T \)
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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

−17.5914679822596342572799289938, −17.12380165645575434732958344171, −16.40696670577639621967160374550, −15.55408325226373186240325998674, −15.12335801716548868025023081419, −14.67644375666387645232055216968, −13.7095796710737127922171633932, −13.13374442032693101585213100708, −11.821637564190995128383264093430, −11.372028909474363228728061906730, −10.88093904020668700362805560256, −10.46100399840805215418971218288, −9.682320352233298530217231712478, −8.852355256023777422686191107712, −8.07818592030461256977291789040, −7.28124474024644405814371762546, −6.932718800841380287242274746853, −6.17884292153677458380044985069, −5.60871335421032477612510175761, −4.68120143768904844653007806929, −4.09463435395596466391913389572, −3.07009859815472262765248662232, −1.91396984264408679902964778683, −1.07447849009582520563804840592, −0.29201931710767203289707066455, 1.04995345236412784454971413380, 1.54064981285007602907851558429, 2.08634126891339801277977718984, 3.486384541613281397921921429091, 4.2417129010916967405992674627, 4.826679156620415214968991134811, 5.512858309217276675743211616433, 6.59928803616484058695433423695, 7.11400899155768904557762274911, 7.88715664105320530636204324537, 8.732251533953423877768852939640, 9.16108456101775713213416335676, 9.74689095121175520735505003047, 10.859800110193428317235194336423, 11.29135852202492233194230871207, 11.895038437282387062043681911345, 12.21230002246119162766122886202, 12.876275187329785309623044043743, 13.710129890285126371027306440957, 14.592885644679261492805213206135, 15.719225260983572011260033933605, 16.01176403285198923355999782625, 16.737186324458082743548414615029, 17.39844530816004292693858543461, 17.614274447781516147626677322295

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