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Properties

Label 1-6017-6017.1627-r0-0-0
Degree $1$
Conductor $6017$
Sign $0.914 - 0.404i$
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  + (0.365 + 0.930i)2-s + (0.222 + 0.974i)3-s + (−0.733 + 0.680i)4-s + (−0.826 − 0.563i)5-s + (−0.826 + 0.563i)6-s + (−0.733 − 0.680i)7-s + (−0.900 − 0.433i)8-s + (−0.900 + 0.433i)9-s + (0.222 − 0.974i)10-s + (−0.826 − 0.563i)12-s + (0.988 − 0.149i)13-s + (0.365 − 0.930i)14-s + (0.365 − 0.930i)15-s + (0.0747 − 0.997i)16-s + (0.0747 − 0.997i)17-s + (−0.733 − 0.680i)18-s + ⋯
L(s)  = 1  + (0.365 + 0.930i)2-s + (0.222 + 0.974i)3-s + (−0.733 + 0.680i)4-s + (−0.826 − 0.563i)5-s + (−0.826 + 0.563i)6-s + (−0.733 − 0.680i)7-s + (−0.900 − 0.433i)8-s + (−0.900 + 0.433i)9-s + (0.222 − 0.974i)10-s + (−0.826 − 0.563i)12-s + (0.988 − 0.149i)13-s + (0.365 − 0.930i)14-s + (0.365 − 0.930i)15-s + (0.0747 − 0.997i)16-s + (0.0747 − 0.997i)17-s + (−0.733 − 0.680i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.914 - 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.914 - 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $0.914 - 0.404i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (1627, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ 0.914 - 0.404i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3114389397 - 0.06582107455i\)
\(L(\frac12)\) \(\approx\) \(0.3114389397 - 0.06582107455i\)
\(L(1)\) \(\approx\) \(0.6368092334 + 0.5319142623i\)
\(L(1)\) \(\approx\) \(0.6368092334 + 0.5319142623i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (0.365 + 0.930i)T \)
3 \( 1 + (0.222 + 0.974i)T \)
5 \( 1 + (-0.826 - 0.563i)T \)
7 \( 1 + (-0.733 - 0.680i)T \)
13 \( 1 + (0.988 - 0.149i)T \)
17 \( 1 + (0.0747 - 0.997i)T \)
19 \( 1 + (-0.365 + 0.930i)T \)
23 \( 1 + (-0.365 + 0.930i)T \)
29 \( 1 + (0.900 + 0.433i)T \)
31 \( 1 + (0.222 + 0.974i)T \)
37 \( 1 + (-0.0747 + 0.997i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.733 - 0.680i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (-0.988 + 0.149i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (0.0747 - 0.997i)T \)
67 \( 1 + (0.0747 - 0.997i)T \)
71 \( 1 + (-0.0747 + 0.997i)T \)
73 \( 1 + (-0.365 - 0.930i)T \)
79 \( 1 + (-0.222 + 0.974i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (0.900 + 0.433i)T \)
97 \( 1 + (0.365 - 0.930i)T \)
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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

−17.99141982115534835060733892893, −17.51097238236671472890245452211, −16.32495057262821092968035579066, −15.590153712214168929025598438461, −14.92702769785649945407342847096, −14.44002542146352808365605574566, −13.57410042800807769051020877409, −13.03061698087028776275527431546, −12.51542099283878256050369407490, −11.85502450757176923755805309299, −11.36442120055077547513018316407, −10.667809831774591816246263635701, −9.95600200784820517179517391916, −8.84839155348328076046907474745, −8.61590525881493089030869014366, −7.83496511105354512199118271078, −6.749519581150847897416939638327, −6.25814928997744074247066867498, −5.765742720900504448778738648466, −4.50147064701135852993675773821, −3.84562436173457576041446968300, −3.08989542267320555783682634376, −2.576101013748515534165506280893, −1.85693284320185541981704519032, −0.80015625144575153380991808139, 0.0924754342350888184477399912, 1.281957466846512018615989553231, 3.01473827412121334568310772111, 3.52648546166881247001510686116, 3.88546006486693605733618547425, 4.86357379911087395607378551673, 5.14593163074808162046004514257, 6.22178963961249351005996919759, 6.79190314706164658750250433770, 7.746531557440468948055339211620, 8.286032998133555647532269316901, 8.78869344680703080297366556176, 9.65867816243019634732697486477, 10.1137448376159725211030796937, 11.09912262653852489163522955269, 11.81909077860997116962552578004, 12.493423089105211612239618374777, 13.36236553450221256469576372944, 13.803235324992928760301271228331, 14.45226885404494654524407498497, 15.33227118733942399852564631115, 15.83508608914523495000702364210, 16.130773808148898268384852444671, 16.739515746017334378935871602, 17.25160896794837424335242168541

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