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Properties

Label 1-6008-6008.13-r0-0-0
Degree $1$
Conductor $6008$
Sign $0.998 + 0.0613i$
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  + (0.440 + 0.897i)3-s + (0.994 + 0.108i)5-s + (0.938 − 0.344i)7-s + (−0.611 + 0.791i)9-s + (0.268 − 0.963i)11-s + (−0.859 − 0.510i)13-s + (0.340 + 0.940i)15-s + (0.549 − 0.835i)17-s + (0.987 + 0.158i)19-s + (0.723 + 0.690i)21-s + (−0.542 + 0.839i)23-s + (0.976 + 0.216i)25-s + (−0.979 − 0.199i)27-s + (−0.433 + 0.901i)29-s + (0.959 − 0.281i)31-s + ⋯
L(s)  = 1  + (0.440 + 0.897i)3-s + (0.994 + 0.108i)5-s + (0.938 − 0.344i)7-s + (−0.611 + 0.791i)9-s + (0.268 − 0.963i)11-s + (−0.859 − 0.510i)13-s + (0.340 + 0.940i)15-s + (0.549 − 0.835i)17-s + (0.987 + 0.158i)19-s + (0.723 + 0.690i)21-s + (−0.542 + 0.839i)23-s + (0.976 + 0.216i)25-s + (−0.979 − 0.199i)27-s + (−0.433 + 0.901i)29-s + (0.959 − 0.281i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0613i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0613i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6008\)    =    \(2^{3} \cdot 751\)
Sign: $0.998 + 0.0613i$
Analytic conductor: \(27.9010\)
Root analytic conductor: \(27.9010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6008} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6008,\ (0:\ ),\ 0.998 + 0.0613i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.160105994 + 0.09699085904i\)
\(L(\frac12)\) \(\approx\) \(3.160105994 + 0.09699085904i\)
\(L(1)\) \(\approx\) \(1.642332317 + 0.2325096373i\)
\(L(1)\) \(\approx\) \(1.642332317 + 0.2325096373i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
751 \( 1 \)
good3 \( 1 + (0.440 + 0.897i)T \)
5 \( 1 + (0.994 + 0.108i)T \)
7 \( 1 + (0.938 - 0.344i)T \)
11 \( 1 + (0.268 - 0.963i)T \)
13 \( 1 + (-0.859 - 0.510i)T \)
17 \( 1 + (0.549 - 0.835i)T \)
19 \( 1 + (0.987 + 0.158i)T \)
23 \( 1 + (-0.542 + 0.839i)T \)
29 \( 1 + (-0.433 + 0.901i)T \)
31 \( 1 + (0.959 - 0.281i)T \)
37 \( 1 + (-0.0961 - 0.995i)T \)
41 \( 1 + (0.968 - 0.248i)T \)
43 \( 1 + (-0.212 - 0.977i)T \)
47 \( 1 + (0.868 - 0.496i)T \)
53 \( 1 + (0.637 + 0.770i)T \)
59 \( 1 + (-0.999 + 0.0167i)T \)
61 \( 1 + (0.348 - 0.937i)T \)
67 \( 1 + (0.880 - 0.474i)T \)
71 \( 1 + (-0.470 - 0.882i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (-0.987 + 0.158i)T \)
83 \( 1 + (-0.387 - 0.921i)T \)
89 \( 1 + (-0.952 + 0.305i)T \)
97 \( 1 + (-0.170 - 0.985i)T \)
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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

−17.804182696759266565363609565304, −17.20707469950274324115213993048, −16.75615885010894420424298515629, −15.54510050938360529151921016444, −14.745753203586014875873050889296, −14.43078050947631461320894391473, −13.8924072030582503379544253215, −13.11511687647619479474666886192, −12.41682990073322891551040518771, −11.972505393559454765175278890792, −11.3264028829040928187373444082, −10.134810938814231947727187380326, −9.75218796629222425043617733415, −8.994508855426602794225465103582, −8.28684278037215426139389613252, −7.65921776027620085009871849325, −6.989535235361497478890504112805, −6.22759776507264127460334448012, −5.62907194886306960337712612479, −4.77162050967367362376162894784, −4.131952928463845313420356325045, −2.75628681367163563322049844787, −2.39122679821846197682029069547, −1.57724147319246098530186276801, −1.12272424271504123409728159898, 0.75593844381745567460565542473, 1.71109949533828728205442634493, 2.570464701105392551933591427926, 3.19406911854293660999029082285, 3.97240097908377911559245858721, 4.89824876015580084087563415490, 5.49101503492655370341390458301, 5.798197770123487461101152164292, 7.217770252403266683374835789211, 7.62476819486144209605135450154, 8.511000236108351324091241077339, 9.17191400779556252300921148503, 9.75555590631858050722741363271, 10.33550334361493371138154572145, 10.98213568666895477697868183414, 11.60877201940943650132877444633, 12.405102151985765571678562996398, 13.53527618681454447155814996943, 14.071327457941943506325229205129, 14.16499037055791826136605811418, 15.016506519746781889047952257525, 15.76936872556980519087650104158, 16.446987072325289433334127409956, 17.098390454814013846779889820221, 17.523161997713954391591073303169

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