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Properties

Label 1-6008-6008.45-r0-0-0
Degree $1$
Conductor $6008$
Sign $0.800 - 0.599i$
Analytic cond. $27.9010$
Root an. cond. $27.9010$
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  + (−0.994 + 0.100i)3-s + (−0.162 − 0.986i)5-s + (−0.863 − 0.503i)7-s + (0.979 − 0.199i)9-s + (0.929 + 0.368i)11-s + (−0.693 + 0.720i)13-s + (0.260 + 0.965i)15-s + (−0.0878 − 0.996i)17-s + (0.236 + 0.971i)19-s + (0.910 + 0.414i)21-s + (−0.997 + 0.0753i)23-s + (−0.947 + 0.320i)25-s + (−0.954 + 0.297i)27-s + (−0.112 + 0.993i)29-s + (−0.910 − 0.414i)31-s + ⋯
L(s)  = 1  + (−0.994 + 0.100i)3-s + (−0.162 − 0.986i)5-s + (−0.863 − 0.503i)7-s + (0.979 − 0.199i)9-s + (0.929 + 0.368i)11-s + (−0.693 + 0.720i)13-s + (0.260 + 0.965i)15-s + (−0.0878 − 0.996i)17-s + (0.236 + 0.971i)19-s + (0.910 + 0.414i)21-s + (−0.997 + 0.0753i)23-s + (−0.947 + 0.320i)25-s + (−0.954 + 0.297i)27-s + (−0.112 + 0.993i)29-s + (−0.910 − 0.414i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.800 - 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.800 - 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6008\)    =    \(2^{3} \cdot 751\)
Sign: $0.800 - 0.599i$
Analytic conductor: \(27.9010\)
Root analytic conductor: \(27.9010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6008} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6008,\ (0:\ ),\ 0.800 - 0.599i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6194078643 - 0.2060856333i\)
\(L(\frac12)\) \(\approx\) \(0.6194078643 - 0.2060856333i\)
\(L(1)\) \(\approx\) \(0.6180504619 - 0.09273215184i\)
\(L(1)\) \(\approx\) \(0.6180504619 - 0.09273215184i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
751 \( 1 \)
good3 \( 1 + (-0.994 + 0.100i)T \)
5 \( 1 + (-0.162 - 0.986i)T \)
7 \( 1 + (-0.863 - 0.503i)T \)
11 \( 1 + (0.929 + 0.368i)T \)
13 \( 1 + (-0.693 + 0.720i)T \)
17 \( 1 + (-0.0878 - 0.996i)T \)
19 \( 1 + (0.236 + 0.971i)T \)
23 \( 1 + (-0.997 + 0.0753i)T \)
29 \( 1 + (-0.112 + 0.993i)T \)
31 \( 1 + (-0.910 - 0.414i)T \)
37 \( 1 + (0.597 + 0.801i)T \)
41 \( 1 + (-0.929 - 0.368i)T \)
43 \( 1 + (-0.448 - 0.893i)T \)
47 \( 1 + (-0.711 - 0.702i)T \)
53 \( 1 + (-0.968 - 0.248i)T \)
59 \( 1 + (0.999 + 0.0251i)T \)
61 \( 1 + (-0.968 - 0.248i)T \)
67 \( 1 + (0.675 - 0.737i)T \)
71 \( 1 + (0.998 - 0.0502i)T \)
73 \( 1 + T \)
79 \( 1 + (-0.236 + 0.971i)T \)
83 \( 1 + (0.187 + 0.982i)T \)
89 \( 1 + (0.448 - 0.893i)T \)
97 \( 1 + (-0.863 + 0.503i)T \)
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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

−17.71489858068932852047305027929, −17.279141020265374028639215964877, −16.43137780581891052953256384055, −15.894983427591850392035382025533, −15.16921052360495515356472981893, −14.72234183234284497241217715306, −13.798235325503674032477792588, −12.988635320877265411205832242181, −12.4912301387255800088138642434, −11.7384046788013591076770229710, −11.25793850470518614917994164785, −10.57338962805305990145522680601, −9.83814763194634191761489676071, −9.4496080896879721903839591500, −8.30112086218347055611618728938, −7.5393823345629273183214347689, −6.81130878220298152654147663797, −6.21008159831752083788020378856, −5.90003394613696402423354515234, −4.93001161047074319179912138439, −3.99445723392215803336158341523, −3.38561171327347130118533890376, −2.52620703573622683936673538630, −1.69836161872260485281965646825, −0.449408878677870700985254826299, 0.41029448470251765034178360398, 1.384323320406812160712243445949, 2.02854388268079109917413559365, 3.602061668047495396453816352359, 3.90383506914824051999640015695, 4.82501401466793865120495955673, 5.2609033082929268576973727202, 6.2063690789127331796284213735, 6.8147290280792128996080399855, 7.35154299601496314942997716115, 8.256174152853818554420561280058, 9.44629752177656226409509733310, 9.508750430861182091239492137969, 10.20726363135398197751293166110, 11.211358325318607080510723778478, 11.91007531776820181273622642464, 12.22530256809811294459373117983, 12.83750437861529761623350979582, 13.649675266307362430097049251933, 14.26675603985701965237789502533, 15.22709527106069949541661125194, 15.990326283849089733608529798144, 16.56193407993817155492519724166, 16.77455474019855510207505852668, 17.363986363428940518145936060737

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