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Properties

Label 1-6017-6017.5707-r0-0-0
Degree $1$
Conductor $6017$
Sign $-0.998 - 0.0546i$
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  + (0.995 − 0.0965i)2-s + (0.309 − 0.951i)3-s + (0.981 − 0.192i)4-s + (0.0241 + 0.999i)5-s + (0.215 − 0.976i)6-s + (0.399 − 0.916i)7-s + (0.958 − 0.285i)8-s + (−0.809 − 0.587i)9-s + (0.120 + 0.992i)10-s + (0.120 − 0.992i)12-s + (−0.809 − 0.587i)13-s + (0.309 − 0.951i)14-s + (0.958 + 0.285i)15-s + (0.926 − 0.377i)16-s + (−0.989 + 0.144i)17-s + (−0.861 − 0.506i)18-s + ⋯
L(s)  = 1  + (0.995 − 0.0965i)2-s + (0.309 − 0.951i)3-s + (0.981 − 0.192i)4-s + (0.0241 + 0.999i)5-s + (0.215 − 0.976i)6-s + (0.399 − 0.916i)7-s + (0.958 − 0.285i)8-s + (−0.809 − 0.587i)9-s + (0.120 + 0.992i)10-s + (0.120 − 0.992i)12-s + (−0.809 − 0.587i)13-s + (0.309 − 0.951i)14-s + (0.958 + 0.285i)15-s + (0.926 − 0.377i)16-s + (−0.989 + 0.144i)17-s + (−0.861 − 0.506i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0546i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0546i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $-0.998 - 0.0546i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (5707, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ -0.998 - 0.0546i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05769843458 - 2.111908697i\)
\(L(\frac12)\) \(\approx\) \(0.05769843458 - 2.111908697i\)
\(L(1)\) \(\approx\) \(1.590966829 - 0.8081390938i\)
\(L(1)\) \(\approx\) \(1.590966829 - 0.8081390938i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (0.995 - 0.0965i)T \)
3 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 + (0.0241 + 0.999i)T \)
7 \( 1 + (0.399 - 0.916i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
17 \( 1 + (-0.989 + 0.144i)T \)
19 \( 1 + (-0.0724 + 0.997i)T \)
23 \( 1 + (-0.354 - 0.935i)T \)
29 \( 1 + (-0.998 - 0.0483i)T \)
31 \( 1 + (0.644 + 0.764i)T \)
37 \( 1 + (-0.998 - 0.0483i)T \)
41 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 + (0.120 - 0.992i)T \)
47 \( 1 + (0.779 + 0.626i)T \)
53 \( 1 + (-0.943 + 0.331i)T \)
59 \( 1 + (-0.0724 - 0.997i)T \)
61 \( 1 + (0.215 - 0.976i)T \)
67 \( 1 + (0.120 - 0.992i)T \)
71 \( 1 + (0.644 - 0.764i)T \)
73 \( 1 + (-0.607 + 0.794i)T \)
79 \( 1 + (-0.681 + 0.732i)T \)
83 \( 1 + (0.0241 + 0.999i)T \)
89 \( 1 + (-0.748 - 0.663i)T \)
97 \( 1 + (-0.443 - 0.896i)T \)
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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

−17.53408517956598957193400630650, −17.3595099444414582478094585181, −16.354943810154766528371296113306, −16.00047783621910418735164066865, −15.23185887699720770188352775985, −14.99755924736040801972675647977, −14.13698651864694304559047673318, −13.425254891242170612639957912680, −12.98124318767649587730621855698, −11.9566185313918123230447837815, −11.60625367104812392102985863027, −11.06061670709326825498182282033, −9.98702245678475808740207859583, −9.293548330515107463555585259117, −8.82003040489159163481564992404, −8.02787478083177839338779468591, −7.31511982054818462498777876137, −6.25411007466477043690361328253, −5.51646922196090343276469988624, −5.011728043443349276402481571619, −4.44831304210178682975300806040, −3.94626616814332655121817873567, −2.80141482705068414898535155349, −2.315456435224211027312803428266, −1.530811176475999230530817738138, 0.2801616204072727533652251204, 1.58321920115109887581231373559, 2.12591715055344889449170708466, 2.83543198176051212772899658085, 3.60282916051502134014293903916, 4.18874399966135901494268550014, 5.20510103987640406977505235657, 5.95622959299389488100742662399, 6.73482431300881129401760914498, 7.06013627889939047133625306295, 7.77408611029827717600838819645, 8.29750332031703128122771179673, 9.5801674495453822709426533250, 10.49929628106208483327026429778, 10.78899393545426212962343419573, 11.54933852540385998644105557651, 12.40943412888408877606221324584, 12.69000833753339609038284212114, 13.67235990992125010985616396071, 14.18006322028189669102960676427, 14.35373997455230736211427537171, 15.23898479693424115741984690342, 15.73489138141980995340717548822, 17.00640454004648574266571500426, 17.26207208429994036748553805235

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