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Properties

Label 1-6001-6001.4611-r0-0-0
Degree $1$
Conductor $6001$
Sign $0.980 + 0.198i$
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  + (0.654 − 0.755i)2-s + (−0.755 + 0.654i)3-s + (−0.142 − 0.989i)4-s + (−0.755 + 0.654i)5-s + i·6-s + i·7-s + (−0.841 − 0.540i)8-s + (0.142 − 0.989i)9-s + i·10-s + (−0.909 + 0.415i)11-s + (0.755 + 0.654i)12-s + (−0.959 + 0.281i)13-s + (0.755 + 0.654i)14-s + (0.142 − 0.989i)15-s + (−0.959 + 0.281i)16-s + ⋯
L(s)  = 1  + (0.654 − 0.755i)2-s + (−0.755 + 0.654i)3-s + (−0.142 − 0.989i)4-s + (−0.755 + 0.654i)5-s + i·6-s + i·7-s + (−0.841 − 0.540i)8-s + (0.142 − 0.989i)9-s + i·10-s + (−0.909 + 0.415i)11-s + (0.755 + 0.654i)12-s + (−0.959 + 0.281i)13-s + (0.755 + 0.654i)14-s + (0.142 − 0.989i)15-s + (−0.959 + 0.281i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.198i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.198i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6001\)    =    \(17 \cdot 353\)
Sign: $0.980 + 0.198i$
Analytic conductor: \(27.8685\)
Root analytic conductor: \(27.8685\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6001} (4611, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6001,\ (0:\ ),\ 0.980 + 0.198i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.094927046 + 0.1096808653i\)
\(L(\frac12)\) \(\approx\) \(1.094927046 + 0.1096808653i\)
\(L(1)\) \(\approx\) \(0.8744476400 - 0.04769711145i\)
\(L(1)\) \(\approx\) \(0.8744476400 - 0.04769711145i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
353 \( 1 \)
good2 \( 1 + (0.654 - 0.755i)T \)
3 \( 1 + (-0.755 + 0.654i)T \)
5 \( 1 + (-0.755 + 0.654i)T \)
7 \( 1 + iT \)
11 \( 1 + (-0.909 + 0.415i)T \)
13 \( 1 + (-0.959 + 0.281i)T \)
19 \( 1 + (0.959 + 0.281i)T \)
23 \( 1 + (0.909 + 0.415i)T \)
29 \( 1 + (0.989 + 0.142i)T \)
31 \( 1 + (0.909 - 0.415i)T \)
37 \( 1 + (0.540 - 0.841i)T \)
41 \( 1 + (-0.909 + 0.415i)T \)
43 \( 1 + (-0.415 - 0.909i)T \)
47 \( 1 + (-0.142 - 0.989i)T \)
53 \( 1 + (0.654 + 0.755i)T \)
59 \( 1 - T \)
61 \( 1 + (-0.281 - 0.959i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.989 + 0.142i)T \)
73 \( 1 + (-0.989 + 0.142i)T \)
79 \( 1 + (-0.755 - 0.654i)T \)
83 \( 1 + (-0.841 + 0.540i)T \)
89 \( 1 + (0.841 - 0.540i)T \)
97 \( 1 + (-0.540 - 0.841i)T \)
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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

−17.46371146776243638349715100775, −16.93776854319186238852553546006, −16.38322637585933418997494950930, −15.88842292836379112806767834212, −15.22674561615040616545443132938, −14.320614803627150577423571611863, −13.56947548376848780342012991597, −13.153361555163288898376188431365, −12.588122782735939661766204751305, −11.86432425322289706080267129529, −11.42225458277999357116688561123, −10.55819116880230733716991126700, −9.79675037803005018594868766403, −8.56386660846001296818810828492, −8.05733900950833942165291254667, −7.44037366053765368318927262219, −7.03707887960550036266337052898, −6.24189025267220547539925395664, −5.323227227884862320398536837990, −4.75532359145493496092366587674, −4.50095129337375061991202273539, −3.20426599711976236592714151121, −2.739966146729894245350342942522, −1.2358075879120737672858144588, −0.48691547331403403539147125050, 0.53168242358498628398452522997, 1.81597008420267613902129816956, 2.84396663049924777374991947484, 3.07239084397992475900639623829, 4.10838566279240965243904263616, 4.833706432689374479524281391652, 5.23649265067421779795065305842, 5.98867818599711937748556430964, 6.79744796480641183493643809980, 7.47834827930071021046815691937, 8.573410574621806862631569936649, 9.395272355929888036384275454708, 10.08709300543502727661660245952, 10.43667821471091205872258964397, 11.40553044488568772932834897500, 11.72623485470570187306329507431, 12.24432038318820466413855411591, 12.83760925532881577008826055288, 13.8410748358121754599156257041, 14.65594444750377820642499913745, 15.15932905917782979616406602518, 15.58430661991836462852302512421, 16.085418534128614550330846023277, 17.13831950276123317116890709597, 17.95499296778010433667621761378

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