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Properties

Label 1-6017-6017.226-r0-0-0
Degree $1$
Conductor $6017$
Sign $-0.584 + 0.811i$
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  + (0.601 + 0.798i)2-s + (−0.134 + 0.990i)3-s + (−0.276 + 0.961i)4-s + (0.628 − 0.777i)5-s + (−0.872 + 0.488i)6-s + (0.106 − 0.994i)7-s + (−0.933 + 0.357i)8-s + (−0.963 − 0.266i)9-s + (0.999 + 0.0345i)10-s + (−0.915 − 0.402i)12-s + (−0.753 − 0.657i)13-s + (0.858 − 0.512i)14-s + (0.686 + 0.727i)15-s + (−0.847 − 0.530i)16-s + (−0.918 + 0.396i)17-s + (−0.367 − 0.930i)18-s + ⋯
L(s)  = 1  + (0.601 + 0.798i)2-s + (−0.134 + 0.990i)3-s + (−0.276 + 0.961i)4-s + (0.628 − 0.777i)5-s + (−0.872 + 0.488i)6-s + (0.106 − 0.994i)7-s + (−0.933 + 0.357i)8-s + (−0.963 − 0.266i)9-s + (0.999 + 0.0345i)10-s + (−0.915 − 0.402i)12-s + (−0.753 − 0.657i)13-s + (0.858 − 0.512i)14-s + (0.686 + 0.727i)15-s + (−0.847 − 0.530i)16-s + (−0.918 + 0.396i)17-s + (−0.367 − 0.930i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $-0.584 + 0.811i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ -0.584 + 0.811i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7239975719 + 1.413513002i\)
\(L(\frac12)\) \(\approx\) \(0.7239975719 + 1.413513002i\)
\(L(1)\) \(\approx\) \(0.9869086850 + 0.6564163107i\)
\(L(1)\) \(\approx\) \(0.9869086850 + 0.6564163107i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (0.601 + 0.798i)T \)
3 \( 1 + (-0.134 + 0.990i)T \)
5 \( 1 + (0.628 - 0.777i)T \)
7 \( 1 + (0.106 - 0.994i)T \)
13 \( 1 + (-0.753 - 0.657i)T \)
17 \( 1 + (-0.918 + 0.396i)T \)
19 \( 1 + (-0.909 - 0.415i)T \)
23 \( 1 + (-0.940 + 0.338i)T \)
29 \( 1 + (0.894 - 0.446i)T \)
31 \( 1 + (-0.411 + 0.911i)T \)
37 \( 1 + (0.235 + 0.971i)T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 + (-0.596 + 0.802i)T \)
47 \( 1 + (0.981 - 0.192i)T \)
53 \( 1 + (-0.127 - 0.991i)T \)
59 \( 1 + (0.607 + 0.794i)T \)
61 \( 1 + (0.282 + 0.959i)T \)
67 \( 1 + (0.915 + 0.402i)T \)
71 \( 1 + (-0.796 + 0.604i)T \)
73 \( 1 + (0.982 - 0.185i)T \)
79 \( 1 + (0.0103 - 0.999i)T \)
83 \( 1 + (0.215 + 0.976i)T \)
89 \( 1 + (0.792 - 0.609i)T \)
97 \( 1 + (0.348 - 0.937i)T \)
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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

−17.67560666262441408705274076562, −17.19964893175344298611278510777, −16.04370633878779928412214968440, −15.2098428567429416682270343184, −14.56125229133036804292628322084, −14.10126481488683098246176391330, −13.5833642599277611203981520219, −12.674923870381977996437362777875, −12.33243913777360592695392094511, −11.66668694826618291363341088240, −11.01135843558961163504617025824, −10.45815191435990438869592108858, −9.47517967194568011105623523683, −9.02096182760295777208243656074, −8.15997661416063356612007395170, −7.09466869282008712940737241700, −6.52038988754135576355363227449, −5.95558319199535682722656375734, −5.37006093560377116191756666018, −4.51493725780706557651132385016, −3.55355265592864287425368029401, −2.508989153953838536608950660719, −2.19117777815602445993755077577, −1.86196416561362432644629286713, −0.41759947028955070666159862916, 0.70023516082278565665986444560, 2.11758855397400849678911086574, 2.95405244416649330259919960147, 3.8801582405822856441767940363, 4.55206338602780794668785849431, 4.79578941656554454217578523133, 5.664765947556173618272923614548, 6.31938782233735124905765248033, 6.980055808628894674316601258197, 8.11331486359915541017024265187, 8.40444777441162887824716076370, 9.25046384098411729018077150582, 10.00138720437346992439025433598, 10.45764186685315332554012184734, 11.42179383838702239256976434209, 12.105554103022407695456462621807, 13.00811430028865325254776141622, 13.345320977882779928467038371257, 14.14682377314221920829641002204, 14.69345057103936114615614817723, 15.34783734905633496426969351387, 16.06014629948678778078405645518, 16.52093715276173668768545351657, 17.2486322564445552318462383102, 17.50712680581377680689751271627

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