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Properties

Label 1-6001-6001.1031-r0-0-0
Degree $1$
Conductor $6001$
Sign $-0.565 + 0.824i$
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  + (0.909 + 0.415i)2-s + (0.584 + 0.811i)3-s + (0.654 + 0.755i)4-s + (−0.987 − 0.159i)5-s + (0.195 + 0.980i)6-s + (−0.831 + 0.555i)7-s + (0.281 + 0.959i)8-s + (−0.315 + 0.948i)9-s + (−0.831 − 0.555i)10-s + (0.570 − 0.821i)11-s + (−0.229 + 0.973i)12-s + (0.943 + 0.332i)13-s + (−0.987 + 0.159i)14-s + (−0.447 − 0.894i)15-s + (−0.142 + 0.989i)16-s + ⋯
L(s)  = 1  + (0.909 + 0.415i)2-s + (0.584 + 0.811i)3-s + (0.654 + 0.755i)4-s + (−0.987 − 0.159i)5-s + (0.195 + 0.980i)6-s + (−0.831 + 0.555i)7-s + (0.281 + 0.959i)8-s + (−0.315 + 0.948i)9-s + (−0.831 − 0.555i)10-s + (0.570 − 0.821i)11-s + (−0.229 + 0.973i)12-s + (0.943 + 0.332i)13-s + (−0.987 + 0.159i)14-s + (−0.447 − 0.894i)15-s + (−0.142 + 0.989i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.565 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.565 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6001\)    =    \(17 \cdot 353\)
Sign: $-0.565 + 0.824i$
Analytic conductor: \(27.8685\)
Root analytic conductor: \(27.8685\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6001} (1031, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6001,\ (0:\ ),\ -0.565 + 0.824i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.759527150 + 3.338103616i\)
\(L(\frac12)\) \(\approx\) \(1.759527150 + 3.338103616i\)
\(L(1)\) \(\approx\) \(1.573718662 + 1.211758088i\)
\(L(1)\) \(\approx\) \(1.573718662 + 1.211758088i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
353 \( 1 \)
good2 \( 1 + (0.909 + 0.415i)T \)
3 \( 1 + (0.584 + 0.811i)T \)
5 \( 1 + (-0.987 - 0.159i)T \)
7 \( 1 + (-0.831 + 0.555i)T \)
11 \( 1 + (0.570 - 0.821i)T \)
13 \( 1 + (0.943 + 0.332i)T \)
19 \( 1 + (0.968 + 0.247i)T \)
23 \( 1 + (0.540 - 0.841i)T \)
29 \( 1 + (0.894 - 0.447i)T \)
31 \( 1 + (-0.0178 + 0.999i)T \)
37 \( 1 + (-0.0891 - 0.996i)T \)
41 \( 1 + (0.540 + 0.841i)T \)
43 \( 1 + (0.984 - 0.177i)T \)
47 \( 1 + (0.315 + 0.948i)T \)
53 \( 1 + (0.811 - 0.584i)T \)
59 \( 1 + (-0.195 - 0.980i)T \)
61 \( 1 + (0.968 - 0.247i)T \)
67 \( 1 + (0.980 + 0.195i)T \)
71 \( 1 + (0.964 + 0.264i)T \)
73 \( 1 + (-0.948 + 0.315i)T \)
79 \( 1 + (0.987 - 0.159i)T \)
83 \( 1 + (-0.959 - 0.281i)T \)
89 \( 1 + (0.886 + 0.463i)T \)
97 \( 1 + (-0.106 - 0.994i)T \)
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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

−17.60385520664083179020313449454, −16.70570773154460343483346913342, −15.7686487642962546754372550727, −15.49597601011612011218437020379, −14.77091859583811411394308733182, −14.05692090258065630832612733337, −13.44501429315108296340997120311, −13.01117680005100551852144192418, −12.19267941431477789847322109005, −11.86317885035666124134431613583, −11.09588869595583319565506230524, −10.33271358449424384405719899641, −9.515997358463959216416336901052, −8.87983641988781778553181823027, −7.79559170583256929808111818294, −7.24682653215219821174114365747, −6.782670714431045167028573732438, −6.10661267292804682705909524190, −5.17650842269470660357195148076, −4.04907638529614260166275863789, −3.77057823853047226175148612000, −3.06086862734676793419164939673, −2.42551132138574125171726021088, −1.194764509950108335068014537690, −0.82635819185716758803594994670, 0.9864635266543304438290482410, 2.403793533541822063579154955183, 3.09993730563752480357883777702, 3.595933880322974818231938431915, 4.103112854309671691631979327969, 4.90194707350818028638542924659, 5.66280343247586721085926499226, 6.37021406901583141635796148365, 7.09154251662493476310492677024, 7.98203614841727201025413785461, 8.63755956222900623954422261274, 8.95170750124910771350351665012, 9.939487523122406825174132844891, 11.06187965474145513843540445454, 11.22427503421205371391449875655, 12.170457061892279063234163456206, 12.6981274823926233113230258276, 13.50628549553447692655153930561, 14.22179559997898779853297972286, 14.5575175173525227437284606285, 15.5586746644564709235088628283, 15.885058717728162610493014269732, 16.253153397037391193845570384354, 16.74309490522803897794117204005, 17.84537889957264832867849301655

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