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Properties

Label 1-6008-6008.397-r0-0-0
Degree $1$
Conductor $6008$
Sign $0.999 + 0.0161i$
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  + (0.818 + 0.574i)3-s + (0.650 + 0.759i)5-s + (0.656 − 0.754i)7-s + (0.340 + 0.940i)9-s + (−0.832 − 0.553i)11-s + (0.941 − 0.336i)13-s + (0.0961 + 0.995i)15-s + (−0.989 − 0.141i)17-s + (−0.859 − 0.510i)19-s + (0.970 − 0.240i)21-s + (0.195 − 0.980i)23-s + (−0.154 + 0.988i)25-s + (−0.260 + 0.965i)27-s + (0.528 + 0.848i)29-s + (0.276 − 0.960i)31-s + ⋯
L(s)  = 1  + (0.818 + 0.574i)3-s + (0.650 + 0.759i)5-s + (0.656 − 0.754i)7-s + (0.340 + 0.940i)9-s + (−0.832 − 0.553i)11-s + (0.941 − 0.336i)13-s + (0.0961 + 0.995i)15-s + (−0.989 − 0.141i)17-s + (−0.859 − 0.510i)19-s + (0.970 − 0.240i)21-s + (0.195 − 0.980i)23-s + (−0.154 + 0.988i)25-s + (−0.260 + 0.965i)27-s + (0.528 + 0.848i)29-s + (0.276 − 0.960i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0161i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0161i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6008\)    =    \(2^{3} \cdot 751\)
Sign: $0.999 + 0.0161i$
Analytic conductor: \(27.9010\)
Root analytic conductor: \(27.9010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6008} (397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6008,\ (0:\ ),\ 0.999 + 0.0161i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.050870835 + 0.02459628325i\)
\(L(\frac12)\) \(\approx\) \(3.050870835 + 0.02459628325i\)
\(L(1)\) \(\approx\) \(1.636128728 + 0.2102597460i\)
\(L(1)\) \(\approx\) \(1.636128728 + 0.2102597460i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
751 \( 1 \)
good3 \( 1 + (0.818 + 0.574i)T \)
5 \( 1 + (0.650 + 0.759i)T \)
7 \( 1 + (0.656 - 0.754i)T \)
11 \( 1 + (-0.832 - 0.553i)T \)
13 \( 1 + (0.941 - 0.336i)T \)
17 \( 1 + (-0.989 - 0.141i)T \)
19 \( 1 + (-0.859 - 0.510i)T \)
23 \( 1 + (0.195 - 0.980i)T \)
29 \( 1 + (0.528 + 0.848i)T \)
31 \( 1 + (0.276 - 0.960i)T \)
37 \( 1 + (0.987 - 0.158i)T \)
41 \( 1 + (0.0627 - 0.998i)T \)
43 \( 1 + (-0.823 - 0.567i)T \)
47 \( 1 + (0.972 + 0.232i)T \)
53 \( 1 + (-0.535 + 0.844i)T \)
59 \( 1 + (0.994 + 0.108i)T \)
61 \( 1 + (0.999 + 0.0418i)T \)
67 \( 1 + (0.0711 - 0.997i)T \)
71 \( 1 + (-0.675 - 0.737i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (0.859 - 0.510i)T \)
83 \( 1 + (-0.228 + 0.973i)T \)
89 \( 1 + (-0.903 - 0.429i)T \)
97 \( 1 + (0.324 - 0.945i)T \)
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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

−17.748341577034306789296818918158, −17.42120299588286092878598301168, −16.23632349553203387131643289993, −15.693238539393592366170226000273, −15.01906432969890894250974585332, −14.45677366799250966322662712646, −13.55740623873488944393376962137, −13.200740841118371253056787804891, −12.6861868291667282752657863852, −11.8831298802174302996648156079, −11.25808087658013023794227406210, −10.22918961673540313890216190653, −9.59010220643860621115542482892, −8.86976048216346566478735639604, −8.31923645576918066427270774268, −8.01975011385952951818049377996, −6.897036311278332750378966429857, −6.22936024825993709625698972153, −5.572511376730987495111777937946, −4.67153218279088332824142860043, −4.132467510868813294927145833443, −2.98026283248142933396728420909, −2.2002209037290733444340907571, −1.7879969969612931442055901785, −1.01341012874327629884749308949, 0.697006060897240297268370619063, 1.91337321112151518608780937503, 2.48063138198131763049175257879, 3.14596121896921832281516097557, 4.010082253605597972551838723056, 4.59976261729377587911297274949, 5.42407025873951235128585290694, 6.26861213568299650628999071308, 7.02033573707819431640606586137, 7.716086276284884113962549377010, 8.56285795814317183150508210198, 8.82278122165689437533537212643, 9.89934172025363404145331157159, 10.57695659336675305739436849210, 10.8243482593641025700725776422, 11.3540026371695955224769166637, 12.89344646295370803498764303838, 13.26051523887168388581778663866, 13.91716073582653395103979150573, 14.296171864773274771376474595512, 15.21320804138240051561723364307, 15.47040975938114657114194820936, 16.396989734929633605405834307195, 17.01920481888773357440031571685, 17.81716038990631511123101644050

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