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Properties

Label 1-6001-6001.2053-r0-0-0
Degree $1$
Conductor $6001$
Sign $-0.222 - 0.974i$
Analytic cond. $27.8685$
Root an. cond. $27.8685$
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  + (−0.989 + 0.142i)2-s + (−0.860 − 0.510i)3-s + (0.959 − 0.281i)4-s + (0.510 − 0.860i)5-s + (0.923 + 0.382i)6-s + (0.923 − 0.382i)7-s + (−0.909 + 0.415i)8-s + (0.479 + 0.877i)9-s + (−0.382 + 0.923i)10-s + (0.654 − 0.755i)11-s + (−0.968 − 0.247i)12-s + (0.984 − 0.177i)13-s + (−0.860 + 0.510i)14-s + (−0.877 + 0.479i)15-s + (0.841 − 0.540i)16-s + ⋯
L(s)  = 1  + (−0.989 + 0.142i)2-s + (−0.860 − 0.510i)3-s + (0.959 − 0.281i)4-s + (0.510 − 0.860i)5-s + (0.923 + 0.382i)6-s + (0.923 − 0.382i)7-s + (−0.909 + 0.415i)8-s + (0.479 + 0.877i)9-s + (−0.382 + 0.923i)10-s + (0.654 − 0.755i)11-s + (−0.968 − 0.247i)12-s + (0.984 − 0.177i)13-s + (−0.860 + 0.510i)14-s + (−0.877 + 0.479i)15-s + (0.841 − 0.540i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.222 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.222 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6001\)    =    \(17 \cdot 353\)
Sign: $-0.222 - 0.974i$
Analytic conductor: \(27.8685\)
Root analytic conductor: \(27.8685\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6001} (2053, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6001,\ (0:\ ),\ -0.222 - 0.974i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8445336080 - 1.059250282i\)
\(L(\frac12)\) \(\approx\) \(0.8445336080 - 1.059250282i\)
\(L(1)\) \(\approx\) \(0.7276124857 - 0.3221617260i\)
\(L(1)\) \(\approx\) \(0.7276124857 - 0.3221617260i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
353 \( 1 \)
good2 \( 1 + (-0.989 + 0.142i)T \)
3 \( 1 + (-0.860 - 0.510i)T \)
5 \( 1 + (0.510 - 0.860i)T \)
7 \( 1 + (0.923 - 0.382i)T \)
11 \( 1 + (0.654 - 0.755i)T \)
13 \( 1 + (0.984 - 0.177i)T \)
19 \( 1 + (0.977 - 0.212i)T \)
23 \( 1 + (0.0713 - 0.997i)T \)
29 \( 1 + (-0.959 - 0.281i)T \)
31 \( 1 + (-0.894 + 0.447i)T \)
37 \( 1 + (0.731 + 0.681i)T \)
41 \( 1 + (0.997 - 0.0713i)T \)
43 \( 1 + (0.0713 + 0.997i)T \)
47 \( 1 + (0.877 + 0.479i)T \)
53 \( 1 + (0.968 - 0.247i)T \)
59 \( 1 + (0.382 + 0.923i)T \)
61 \( 1 + (-0.841 + 0.540i)T \)
67 \( 1 + (0.923 + 0.382i)T \)
71 \( 1 + (-0.627 - 0.778i)T \)
73 \( 1 + (0.959 - 0.281i)T \)
79 \( 1 + (-0.510 - 0.860i)T \)
83 \( 1 + (0.909 + 0.415i)T \)
89 \( 1 + (-0.731 + 0.681i)T \)
97 \( 1 + (0.909 + 0.415i)T \)
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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

−17.98973321309509611140055020026, −17.23452070116190128968369657635, −16.97263144118701455719069724752, −15.92846284493345568389615383952, −15.50132125927869251288562816898, −14.780162817339595791394826549141, −14.28138408794142666301933819385, −13.18586735516275207410452903958, −12.265849070866388888545825071896, −11.62174151438761122988904937151, −11.14418490288570242119846436984, −10.77169778307075930374531217076, −9.88525878522479050574381405971, −9.34250363344023462557146074658, −8.91061748903349684947482625027, −7.63233440687256581937137717420, −7.2897813489539830140089056105, −6.464100257247346482197879074429, −5.717542679780070219290061694082, −5.344520104187505095455545049793, −3.94454019831823283967308460627, −3.588299600749389646228218021142, −2.340144574227930301567551984037, −1.68666763964454226947042859441, −0.99976211984049451501518408717, 0.70611149190880816079236221878, 1.10398310059437996511923397945, 1.68427037490014623406551473417, 2.62866814350264683777543372282, 3.905810144186209862642308639728, 4.77270987749138048754771532303, 5.66938109893144854952311028772, 5.93009036903462558849892467226, 6.77456372435552002823100209193, 7.58263029283887930529914206207, 8.13563289476477684314827473481, 8.83161223616484748015050541547, 9.39727480612562889184522196856, 10.36489014168632990449908185351, 10.94495323452910007772832522369, 11.43746195817835682989276362071, 12.01958559311906265544699117622, 12.84050113620706292746831486014, 13.57507508149407456492892709305, 14.173301440611966381602913707557, 15.036585084344768197031107605651, 16.072524219347156216277379271656, 16.48648263060022792430033584480, 16.84785630408015080075777824322, 17.587995538201465602803816260111

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