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Properties

Label 1-6008-6008.245-r0-0-0
Degree $1$
Conductor $6008$
Sign $0.748 + 0.663i$
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  + (0.203 + 0.979i)3-s + (−0.964 + 0.264i)5-s + (0.994 − 0.100i)7-s + (−0.916 + 0.399i)9-s + (−0.783 + 0.621i)11-s + (0.828 − 0.560i)13-s + (−0.455 − 0.890i)15-s + (0.999 + 0.0167i)17-s + (−0.0293 − 0.999i)19-s + (0.301 + 0.953i)21-s + (−0.723 − 0.690i)23-s + (0.859 − 0.510i)25-s + (−0.577 − 0.816i)27-s + (−0.813 + 0.580i)29-s + (0.976 + 0.216i)31-s + ⋯
L(s)  = 1  + (0.203 + 0.979i)3-s + (−0.964 + 0.264i)5-s + (0.994 − 0.100i)7-s + (−0.916 + 0.399i)9-s + (−0.783 + 0.621i)11-s + (0.828 − 0.560i)13-s + (−0.455 − 0.890i)15-s + (0.999 + 0.0167i)17-s + (−0.0293 − 0.999i)19-s + (0.301 + 0.953i)21-s + (−0.723 − 0.690i)23-s + (0.859 − 0.510i)25-s + (−0.577 − 0.816i)27-s + (−0.813 + 0.580i)29-s + (0.976 + 0.216i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.748 + 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.748 + 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6008\)    =    \(2^{3} \cdot 751\)
Sign: $0.748 + 0.663i$
Analytic conductor: \(27.9010\)
Root analytic conductor: \(27.9010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6008} (245, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6008,\ (0:\ ),\ 0.748 + 0.663i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.444358514 + 0.5483060576i\)
\(L(\frac12)\) \(\approx\) \(1.444358514 + 0.5483060576i\)
\(L(1)\) \(\approx\) \(0.9736392056 + 0.3160205816i\)
\(L(1)\) \(\approx\) \(0.9736392056 + 0.3160205816i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
751 \( 1 \)
good3 \( 1 + (0.203 + 0.979i)T \)
5 \( 1 + (-0.964 + 0.264i)T \)
7 \( 1 + (0.994 - 0.100i)T \)
11 \( 1 + (-0.783 + 0.621i)T \)
13 \( 1 + (0.828 - 0.560i)T \)
17 \( 1 + (0.999 + 0.0167i)T \)
19 \( 1 + (-0.0293 - 0.999i)T \)
23 \( 1 + (-0.723 - 0.690i)T \)
29 \( 1 + (-0.813 + 0.580i)T \)
31 \( 1 + (0.976 + 0.216i)T \)
37 \( 1 + (-0.751 - 0.659i)T \)
41 \( 1 + (-0.929 - 0.368i)T \)
43 \( 1 + (-0.162 + 0.986i)T \)
47 \( 1 + (-0.624 + 0.781i)T \)
53 \( 1 + (-0.968 - 0.248i)T \)
59 \( 1 + (-0.0795 - 0.996i)T \)
61 \( 1 + (0.268 + 0.963i)T \)
67 \( 1 + (0.999 + 0.00837i)T \)
71 \( 1 + (0.356 + 0.934i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (0.0293 - 0.999i)T \)
83 \( 1 + (0.756 - 0.653i)T \)
89 \( 1 + (0.773 - 0.634i)T \)
97 \( 1 + (-0.410 - 0.911i)T \)
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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

−17.79698543168323077547528292507, −16.872232394706452413709604761153, −16.52067169685361672696963425029, −15.50245657594789794565211888068, −15.116541300882820197807138381931, −14.12649295787983858907372735025, −13.80459744577812905761565699461, −13.08841262212037398684238291685, −12.12610033453275935524322810295, −11.86810463364264483127485391436, −11.27119559781122642478328748572, −10.54698041625061610864324444042, −9.55042898457931469398218961330, −8.47288298735053887778129631456, −8.210056577883820865956028874892, −7.812699951783045524834666689916, −7.04806791285656698798160107229, −6.093898657535308087403857967738, −5.507764191451726025924060922058, −4.74340026952113684431226395394, −3.61310084477027906103523963034, −3.36745111562094793743502690028, −2.089458197239020885877152627769, −1.51499708162499215923502066063, −0.679131092473247199420953578928, 0.565121924632363785809563069426, 1.7903444737057475435258377720, 2.77674553708091882870069040740, 3.38329929532033503182111309009, 4.11248368402353254067628828997, 4.85643228175506662142623000357, 5.21202606880511600755965488270, 6.22790408056465450528400995764, 7.26569180100130418187257260140, 7.93748589676362728482195165171, 8.31472842046361716425165325200, 9.00802775794861178554436101504, 10.10069498328970849687406099367, 10.44813529606294868594721824496, 11.18468579603265870873462247824, 11.57640186623992197693733987437, 12.46353404917087416063841945569, 13.19652638805230217974609605716, 14.26198361301869204250207799229, 14.508414250149292914928869608899, 15.29882755244098646065657357012, 15.76654241133850076483764808803, 16.18689774516251089305088270330, 17.142287707457545376596954751488, 17.757607659546174452695237128924

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