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Properties

Label 1-6017-6017.1316-r0-0-0
Degree $1$
Conductor $6017$
Sign $0.0571 + 0.998i$
Analytic cond. $27.9428$
Root an. cond. $27.9428$
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  + (−0.726 + 0.686i)2-s + (0.809 + 0.587i)3-s + (0.0563 − 0.998i)4-s + (−0.899 + 0.435i)5-s + (−0.991 + 0.128i)6-s + (0.759 − 0.650i)7-s + (0.644 + 0.764i)8-s + (0.309 + 0.951i)9-s + (0.354 − 0.935i)10-s + (0.632 − 0.774i)12-s + (0.978 + 0.207i)13-s + (−0.104 + 0.994i)14-s + (−0.984 − 0.176i)15-s + (−0.993 − 0.112i)16-s + (0.818 − 0.574i)17-s + (−0.877 − 0.478i)18-s + ⋯
L(s)  = 1  + (−0.726 + 0.686i)2-s + (0.809 + 0.587i)3-s + (0.0563 − 0.998i)4-s + (−0.899 + 0.435i)5-s + (−0.991 + 0.128i)6-s + (0.759 − 0.650i)7-s + (0.644 + 0.764i)8-s + (0.309 + 0.951i)9-s + (0.354 − 0.935i)10-s + (0.632 − 0.774i)12-s + (0.978 + 0.207i)13-s + (−0.104 + 0.994i)14-s + (−0.984 − 0.176i)15-s + (−0.993 − 0.112i)16-s + (0.818 − 0.574i)17-s + (−0.877 − 0.478i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0571 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0571 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $0.0571 + 0.998i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (1316, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ 0.0571 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.431035595 + 1.351427495i\)
\(L(\frac12)\) \(\approx\) \(1.431035595 + 1.351427495i\)
\(L(1)\) \(\approx\) \(0.9544829758 + 0.5334023190i\)
\(L(1)\) \(\approx\) \(0.9544829758 + 0.5334023190i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (-0.726 + 0.686i)T \)
3 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + (-0.899 + 0.435i)T \)
7 \( 1 + (0.759 - 0.650i)T \)
13 \( 1 + (0.978 + 0.207i)T \)
17 \( 1 + (0.818 - 0.574i)T \)
19 \( 1 + (-0.737 + 0.675i)T \)
23 \( 1 + (0.845 + 0.534i)T \)
29 \( 1 + (0.989 - 0.144i)T \)
31 \( 1 + (0.861 + 0.506i)T \)
37 \( 1 + (-0.369 + 0.929i)T \)
41 \( 1 + (0.913 - 0.406i)T \)
43 \( 1 + (0.987 + 0.160i)T \)
47 \( 1 + (-0.554 + 0.832i)T \)
53 \( 1 + (0.999 - 0.0322i)T \)
59 \( 1 + (-0.737 - 0.675i)T \)
61 \( 1 + (0.991 - 0.128i)T \)
67 \( 1 + (0.987 + 0.160i)T \)
71 \( 1 + (0.00805 + 0.999i)T \)
73 \( 1 + (0.136 - 0.990i)T \)
79 \( 1 + (0.779 - 0.626i)T \)
83 \( 1 + (-0.827 - 0.561i)T \)
89 \( 1 + (-0.568 - 0.822i)T \)
97 \( 1 + (-0.657 - 0.753i)T \)
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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

−17.75003820006103247877925321049, −17.125970362395600992320470611782, −16.27101698185409946957704810761, −15.52625334311956103249251242848, −15.0769355267857812622647963409, −14.24960537775238234208448007805, −13.406044157265281315500132487100, −12.665823712500473075995006631564, −12.38013809221682666047775712769, −11.61195285052889424345597453481, −11.025415022706224651305567484357, −10.336249762389822074643765908, −9.21581071820116970145973453634, −8.806420812473439637966586017609, −8.211793383968870554680545745137, −7.94261759573823383296516542085, −7.06370508762813207743755966914, −6.3133280886200284711647659211, −5.15431817367697118324872849559, −4.175656427775217518073321908238, −3.74079611725123626168613810200, −2.76040958285839745044988455092, −2.29654590020215762322940016088, −1.161595271324417873470040455785, −0.866724720994894835274114595, 0.873437337390659231192953918927, 1.57430951150978512347001233434, 2.69247053108429530254001448591, 3.47388657017587519031592816462, 4.31279695372119548075827036645, 4.757794568374935433342657109461, 5.7220451023986745118062172017, 6.72697623024191562395820675064, 7.29659858931903469516332547588, 8.038856779593428432899030901, 8.31088673403438421515143216134, 8.99358414341962098400442825430, 9.9552673211314764017063634556, 10.41086211191894707915591475482, 11.09237358285819034997579379821, 11.530253044761311710956547252144, 12.686776538451947493384695904437, 13.87284163664380768267015546731, 14.04741705371350249602853641632, 14.70921980535892362887609548602, 15.35554686629927330918586879903, 15.87724358986316343406360889457, 16.40675998806199297092584346510, 17.10740354623004105218987769837, 17.860510298524815459021753951

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