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Properties

Label 1-6013-6013.100-r0-0-0
Degree $1$
Conductor $6013$
Sign $0.678 - 0.735i$
Analytic cond. $27.9242$
Root an. cond. $27.9242$
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  + (−0.996 − 0.0804i)2-s + (0.987 + 0.160i)3-s + (0.987 + 0.160i)4-s + (0.987 − 0.160i)5-s + (−0.970 − 0.239i)6-s + (−0.970 − 0.239i)8-s + (0.948 + 0.316i)9-s + (−0.996 + 0.0804i)10-s + (0.278 − 0.960i)11-s + (0.948 + 0.316i)12-s + 13-s + 15-s + (0.948 + 0.316i)16-s + (−0.0402 + 0.999i)17-s + (−0.919 − 0.391i)18-s + (−0.5 − 0.866i)19-s + ⋯
L(s)  = 1  + (−0.996 − 0.0804i)2-s + (0.987 + 0.160i)3-s + (0.987 + 0.160i)4-s + (0.987 − 0.160i)5-s + (−0.970 − 0.239i)6-s + (−0.970 − 0.239i)8-s + (0.948 + 0.316i)9-s + (−0.996 + 0.0804i)10-s + (0.278 − 0.960i)11-s + (0.948 + 0.316i)12-s + 13-s + 15-s + (0.948 + 0.316i)16-s + (−0.0402 + 0.999i)17-s + (−0.919 − 0.391i)18-s + (−0.5 − 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6013 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.678 - 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6013 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.678 - 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6013\)    =    \(7 \cdot 859\)
Sign: $0.678 - 0.735i$
Analytic conductor: \(27.9242\)
Root analytic conductor: \(27.9242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6013} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6013,\ (0:\ ),\ 0.678 - 0.735i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.226019298 - 0.9750293038i\)
\(L(\frac12)\) \(\approx\) \(2.226019298 - 0.9750293038i\)
\(L(1)\) \(\approx\) \(1.288903297 - 0.1779388227i\)
\(L(1)\) \(\approx\) \(1.288903297 - 0.1779388227i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
859 \( 1 \)
good2 \( 1 + (-0.996 - 0.0804i)T \)
3 \( 1 + (0.987 + 0.160i)T \)
5 \( 1 + (0.987 - 0.160i)T \)
11 \( 1 + (0.278 - 0.960i)T \)
13 \( 1 + T \)
17 \( 1 + (-0.0402 + 0.999i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.428 - 0.903i)T \)
29 \( 1 + (-0.748 - 0.663i)T \)
31 \( 1 + (-0.919 - 0.391i)T \)
37 \( 1 + (0.948 + 0.316i)T \)
41 \( 1 + (-0.354 - 0.935i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.919 + 0.391i)T \)
53 \( 1 + (-0.845 + 0.534i)T \)
59 \( 1 + (0.428 + 0.903i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.200 - 0.979i)T \)
71 \( 1 + (0.885 + 0.464i)T \)
73 \( 1 + (0.692 - 0.721i)T \)
79 \( 1 + (-0.632 + 0.774i)T \)
83 \( 1 + (-0.970 + 0.239i)T \)
89 \( 1 + (-0.0402 - 0.999i)T \)
97 \( 1 + (0.885 - 0.464i)T \)
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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

−18.06382213434623208710062879810, −17.309862478536175185201031977, −16.49026131607860150076798201536, −15.958346485023326326559813902202, −15.119131620012040414004683661545, −14.55285130852477357427154363856, −14.122331143922509537492404020406, −13.01978066193674140454230963318, −12.835117782882060816417935649190, −11.71680445133604712621743396158, −10.98068628183533093659817308643, −10.25209110541476153401868511240, −9.51296297790158340362937534291, −9.32673007029028952051725060703, −8.581270635071574862669405746683, −7.787943197591410607602618577355, −7.15027986244874057808470325318, −6.60385961243122504927760938789, −5.837435656883044053105017176736, −4.97058888834728398783268491884, −3.78516113680114662248166352642, −3.1120519650612761929530798893, −2.27266844108048662635183061899, −1.65584529384976520876427821288, −1.169129025210085516857417675168, 0.72445459220427725549526039757, 1.56736042195278178609283239912, 2.16946424675252900636593339532, 2.924126127808131153385125708356, 3.63517313122844596057330355818, 4.48153252789545975536397609111, 5.74737489255964301856082502159, 6.22974550924886916011254763299, 6.906895678898532008011397649785, 7.88138535723943123244187276716, 8.483242754872099844957920207081, 8.978219466900120374203387067242, 9.40031290449460205913669406194, 10.22411006146711045064308408867, 10.921328973147172919415003268048, 11.21222784229830188752385155850, 12.67496168482516225401301922237, 12.92726697476974800124115857801, 13.748972994088281835752246792403, 14.38490538564489067821596014622, 15.132185889968221250680916417534, 15.66438170518771525681508080346, 16.5794991171819522440759763111, 16.87083962366233354758769287028, 17.66207265253521983571845354491

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