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Properties

Label 1-6017-6017.4090-r0-0-0
Degree $1$
Conductor $6017$
Sign $-0.917 - 0.397i$
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  + (0.0241 − 0.999i)2-s + (0.309 − 0.951i)3-s + (−0.998 − 0.0483i)4-s + (0.926 − 0.377i)5-s + (−0.943 − 0.331i)6-s + (0.958 + 0.285i)7-s + (−0.0724 + 0.997i)8-s + (−0.809 − 0.587i)9-s + (−0.354 − 0.935i)10-s + (−0.354 + 0.935i)12-s + (−0.809 − 0.587i)13-s + (0.309 − 0.951i)14-s + (−0.0724 − 0.997i)15-s + (0.995 + 0.0965i)16-s + (−0.681 − 0.732i)17-s + (−0.607 + 0.794i)18-s + ⋯
L(s)  = 1  + (0.0241 − 0.999i)2-s + (0.309 − 0.951i)3-s + (−0.998 − 0.0483i)4-s + (0.926 − 0.377i)5-s + (−0.943 − 0.331i)6-s + (0.958 + 0.285i)7-s + (−0.0724 + 0.997i)8-s + (−0.809 − 0.587i)9-s + (−0.354 − 0.935i)10-s + (−0.354 + 0.935i)12-s + (−0.809 − 0.587i)13-s + (0.309 − 0.951i)14-s + (−0.0724 − 0.997i)15-s + (0.995 + 0.0965i)16-s + (−0.681 − 0.732i)17-s + (−0.607 + 0.794i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.917 - 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.917 - 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $-0.917 - 0.397i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (4090, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ -0.917 - 0.397i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4960914917 - 2.395830450i\)
\(L(\frac12)\) \(\approx\) \(0.4960914917 - 2.395830450i\)
\(L(1)\) \(\approx\) \(0.8870210700 - 1.050334149i\)
\(L(1)\) \(\approx\) \(0.8870210700 - 1.050334149i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (0.0241 - 0.999i)T \)
3 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 + (0.926 - 0.377i)T \)
7 \( 1 + (0.958 + 0.285i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
17 \( 1 + (-0.681 - 0.732i)T \)
19 \( 1 + (0.399 + 0.916i)T \)
23 \( 1 + (0.885 + 0.464i)T \)
29 \( 1 + (0.715 + 0.698i)T \)
31 \( 1 + (0.215 + 0.976i)T \)
37 \( 1 + (0.715 + 0.698i)T \)
41 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 + (-0.354 + 0.935i)T \)
47 \( 1 + (-0.168 - 0.985i)T \)
53 \( 1 + (0.644 + 0.764i)T \)
59 \( 1 + (0.399 - 0.916i)T \)
61 \( 1 + (-0.943 - 0.331i)T \)
67 \( 1 + (-0.354 + 0.935i)T \)
71 \( 1 + (0.215 - 0.976i)T \)
73 \( 1 + (-0.527 - 0.849i)T \)
79 \( 1 + (0.836 - 0.548i)T \)
83 \( 1 + (0.926 - 0.377i)T \)
89 \( 1 + (0.568 - 0.822i)T \)
97 \( 1 + (0.485 - 0.873i)T \)
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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

−17.66632409019073291828279932230, −17.22052501329766847956569173145, −16.81396146056071769077584281356, −16.02038879133524123712745213484, −15.11741068895405583759460911130, −14.8277243080554863422581084331, −14.35863287042364925606898213106, −13.483121839073286794262530942769, −13.33793843462807260785991330449, −12.05708215508804886418804692287, −11.105470538140975224896922904490, −10.632021477050747401743126851379, −9.769519212923831757018471174414, −9.34868826358822186785919942542, −8.68509945339976038205125526041, −7.99583755769018121837537893124, −7.21857738468110812596624991272, −6.51685809250607778968888606099, −5.75482369035406320072635549598, −4.99253894275879984955931981211, −4.555870974582396664068610305916, −3.93289042514190510317424356811, −2.74753893209436322853549182203, −2.18779052255035828984827802000, −0.91619821225910365271660196144, 0.69432293850104649897058640279, 1.434326768829797488508444685528, 1.95389818881031831731224559642, 2.73860852565142754111118704142, 3.224254462341578260292685799782, 4.58825630317367510299334583693, 5.11464754327218529939960625069, 5.66001810985648724857042729976, 6.62247454370158229662545223608, 7.55840471700218755430466041633, 8.171237398826411530330175559983, 8.9192730556215197395699255946, 9.29616533006874895817493604668, 10.214408217612721693521209383966, 10.822470713278438329654940578443, 11.76354669281413277633797919050, 12.11940074695575067320182623745, 12.77698582176203071143232763581, 13.46119734074914481715437748512, 13.90496472253032115426862610067, 14.566939974882501577954220982465, 15.03702000085809699837720630039, 16.3424054283119646815515532551, 17.327081609360203590674091507023, 17.53573773887116780375102521402

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