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Properties

Label 1-6001-6001.2051-r0-0-0
Degree $1$
Conductor $6001$
Sign $0.993 + 0.114i$
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  − 2-s + (0.980 + 0.195i)3-s + 4-s + (−0.555 − 0.831i)5-s + (−0.980 − 0.195i)6-s + (−0.831 − 0.555i)7-s − 8-s + (0.923 + 0.382i)9-s + (0.555 + 0.831i)10-s + (−0.382 − 0.923i)11-s + (0.980 + 0.195i)12-s + (0.555 − 0.831i)13-s + (0.831 + 0.555i)14-s + (−0.382 − 0.923i)15-s + 16-s + ⋯
L(s)  = 1  − 2-s + (0.980 + 0.195i)3-s + 4-s + (−0.555 − 0.831i)5-s + (−0.980 − 0.195i)6-s + (−0.831 − 0.555i)7-s − 8-s + (0.923 + 0.382i)9-s + (0.555 + 0.831i)10-s + (−0.382 − 0.923i)11-s + (0.980 + 0.195i)12-s + (0.555 − 0.831i)13-s + (0.831 + 0.555i)14-s + (−0.382 − 0.923i)15-s + 16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6001\)    =    \(17 \cdot 353\)
Sign: $0.993 + 0.114i$
Analytic conductor: \(27.8685\)
Root analytic conductor: \(27.8685\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6001} (2051, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6001,\ (0:\ ),\ 0.993 + 0.114i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.228129569 + 0.07049629400i\)
\(L(\frac12)\) \(\approx\) \(1.228129569 + 0.07049629400i\)
\(L(1)\) \(\approx\) \(0.8337054524 - 0.1105561474i\)
\(L(1)\) \(\approx\) \(0.8337054524 - 0.1105561474i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
353 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + (0.980 + 0.195i)T \)
5 \( 1 + (-0.555 - 0.831i)T \)
7 \( 1 + (-0.831 - 0.555i)T \)
11 \( 1 + (-0.382 - 0.923i)T \)
13 \( 1 + (0.555 - 0.831i)T \)
19 \( 1 + (-0.382 - 0.923i)T \)
23 \( 1 + (0.707 + 0.707i)T \)
29 \( 1 + (-0.382 + 0.923i)T \)
31 \( 1 + (0.555 + 0.831i)T \)
37 \( 1 + (0.980 - 0.195i)T \)
41 \( 1 + (0.707 + 0.707i)T \)
43 \( 1 + (-0.923 + 0.382i)T \)
47 \( 1 + (0.382 + 0.923i)T \)
53 \( 1 + (0.831 - 0.555i)T \)
59 \( 1 + (-0.831 + 0.555i)T \)
61 \( 1 + (-0.923 + 0.382i)T \)
67 \( 1 + (0.831 + 0.555i)T \)
71 \( 1 + (-0.555 + 0.831i)T \)
73 \( 1 + (0.923 + 0.382i)T \)
79 \( 1 + (-0.555 + 0.831i)T \)
83 \( 1 - iT \)
89 \( 1 + (0.831 + 0.555i)T \)
97 \( 1 + (0.382 + 0.923i)T \)
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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

−18.12232037616567036206662112902, −16.97350793865823606648171595250, −16.42219811771949414824175898433, −15.46601693134567455911869166932, −15.32302667360471583520389772362, −14.73612169061721985719036173198, −13.8787492976342751610276494883, −13.02507509471967277343975391299, −12.28871629024847400531042645797, −11.82592712636856999996434722672, −10.87695960533089748748572660032, −10.20178340398749110764197888420, −9.65606103878860819091521738579, −9.058656416866486403582498355317, −8.322169665717189089101610240938, −7.735881027651780020216532384674, −7.10339902261933723755335686842, −6.4839938464565031287250717207, −5.973714678873138161291373127770, −4.43497709642138094965047676654, −3.70537495715973786034519309283, −2.99809025860023063353806943706, −2.2696974857386473760441399620, −1.87933482532583207377987658355, −0.480707603975178829140425754378, 0.87060993885875134544168258773, 1.20594626450080649931718372334, 2.59134677913661775425645263494, 3.167705440909410863464949271032, 3.663560443299061379341113201311, 4.67532738446950613424804123023, 5.59078173161898709656329629020, 6.4889777070713614877197141283, 7.3102325000262578300074395202, 7.821583461157270020153850012711, 8.49918640705870021797013521777, 8.983674096766830264734186554719, 9.51498500253882635991400962529, 10.363600590978099816546403580273, 10.89808080543364580518345520402, 11.558294787811914660599618197791, 12.815703925944189917762462921695, 12.92836293121125887026304991117, 13.66175273962129825033444897340, 14.666237185596505876011367662007, 15.50243378790337209179685276465, 15.76531882763789046879241178739, 16.40216489456523256123459497924, 16.87920846408204574936501777440, 17.78831653877861301465144232180

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