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Properties

Label 1-6043-6043.1349-r0-0-0
Degree $1$
Conductor $6043$
Sign $0.397 + 0.917i$
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  + (0.338 + 0.940i)2-s + (0.992 + 0.118i)3-s + (−0.770 + 0.637i)4-s + (−0.930 + 0.366i)5-s + (0.225 + 0.974i)6-s + (−0.840 + 0.542i)7-s + (−0.861 − 0.508i)8-s + (0.972 + 0.234i)9-s + (−0.660 − 0.751i)10-s + (−0.5 − 0.866i)11-s + (−0.840 + 0.542i)12-s + (−0.0493 + 0.998i)13-s + (−0.794 − 0.606i)14-s + (−0.967 + 0.254i)15-s + (0.186 − 0.982i)16-s + (0.996 + 0.0789i)17-s + ⋯
L(s)  = 1  + (0.338 + 0.940i)2-s + (0.992 + 0.118i)3-s + (−0.770 + 0.637i)4-s + (−0.930 + 0.366i)5-s + (0.225 + 0.974i)6-s + (−0.840 + 0.542i)7-s + (−0.861 − 0.508i)8-s + (0.972 + 0.234i)9-s + (−0.660 − 0.751i)10-s + (−0.5 − 0.866i)11-s + (−0.840 + 0.542i)12-s + (−0.0493 + 0.998i)13-s + (−0.794 − 0.606i)14-s + (−0.967 + 0.254i)15-s + (0.186 − 0.982i)16-s + (0.996 + 0.0789i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6043 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.397 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6043 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.397 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6043\)
Sign: $0.397 + 0.917i$
Analytic conductor: \(28.0635\)
Root analytic conductor: \(28.0635\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6043} (1349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6043,\ (0:\ ),\ 0.397 + 0.917i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.454044073 + 0.9547922628i\)
\(L(\frac12)\) \(\approx\) \(1.454044073 + 0.9547922628i\)
\(L(1)\) \(\approx\) \(0.9853294499 + 0.6885089055i\)
\(L(1)\) \(\approx\) \(0.9853294499 + 0.6885089055i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad6043 \( 1 \)
good2 \( 1 + (0.338 + 0.940i)T \)
3 \( 1 + (0.992 + 0.118i)T \)
5 \( 1 + (-0.930 + 0.366i)T \)
7 \( 1 + (-0.840 + 0.542i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.0493 + 0.998i)T \)
17 \( 1 + (0.996 + 0.0789i)T \)
19 \( 1 + (0.922 - 0.384i)T \)
23 \( 1 + (-0.630 + 0.776i)T \)
29 \( 1 + (-0.689 - 0.724i)T \)
31 \( 1 + (-0.990 + 0.137i)T \)
37 \( 1 + (-0.956 - 0.292i)T \)
41 \( 1 + (0.937 - 0.348i)T \)
43 \( 1 + (0.731 + 0.682i)T \)
47 \( 1 + (-0.357 - 0.933i)T \)
53 \( 1 + (0.582 - 0.812i)T \)
59 \( 1 + (-0.566 - 0.823i)T \)
61 \( 1 + (-0.394 - 0.919i)T \)
67 \( 1 + (-0.861 - 0.508i)T \)
71 \( 1 + (0.674 + 0.737i)T \)
73 \( 1 + (0.412 - 0.911i)T \)
79 \( 1 + (0.950 - 0.310i)T \)
83 \( 1 + (0.972 - 0.234i)T \)
89 \( 1 + (-0.660 - 0.751i)T \)
97 \( 1 + (0.961 - 0.273i)T \)
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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

−18.04266880818107954745178412345, −16.82386675759936685508623111652, −16.089139351927574215448030406923, −15.48073999058411260404027311465, −14.82954102717411269746615254787, −14.25491160299246112224142317185, −13.50070680696525051427656360470, −12.819139020212958943744548034066, −12.41989300991176031515775833253, −12.04195153034214729224115875766, −10.67111993005551978264299375840, −10.449235664256766585699875811365, −9.56821552818393974334433981586, −9.184181059312879712960010746276, −8.192664812536252294572263752524, −7.607982316331866125382502541519, −7.14437537434395563130825041439, −5.8633235232243218181051319647, −5.1108539843170028851104976419, −4.256761080234322123622441350747, −3.694640387678512945354333406050, −3.14867353484238628286565786340, −2.54063660814752569001615376809, −1.43626660861133101590124499351, −0.7514402874368081720496569733, 0.47151790242385174906831690932, 2.01709863023466246027314566261, 3.010373393384525962034869088535, 3.531161724737933624698160131428, 3.851904262587346310658304826812, 4.93829264726747847572379084170, 5.66015670634311585739703265620, 6.46427698862576076478215634002, 7.24761465663806807200635099327, 7.71344250409063683086832678307, 8.27093766004483158702085435048, 9.219734187631028372792336777627, 9.3700266905987174935634942065, 10.36211607924768316502392134180, 11.41552374590160029266383516580, 12.08381780798932742617274170487, 12.73969272946453707747117249904, 13.466446271713593233764019003004, 14.06661816213301429443656512941, 14.53337936608847889660316219983, 15.35655019498457586829411732878, 15.7750226258376527553340313381, 16.26114131510836456861984924016, 16.70767550120346908085221817112, 18.060098187786751524768596955600

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