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Properties

Label 1-6008-6008.989-r0-0-0
Degree $1$
Conductor $6008$
Sign $-0.139 - 0.990i$
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  + (−0.643 + 0.765i)3-s + (−0.932 + 0.360i)5-s + (−0.137 − 0.990i)7-s + (−0.170 − 0.985i)9-s + (−0.604 + 0.796i)11-s + (0.935 + 0.352i)13-s + (0.324 − 0.945i)15-s + (−0.723 + 0.690i)17-s + (−0.521 + 0.853i)19-s + (0.846 + 0.532i)21-s + (0.129 + 0.991i)23-s + (0.740 − 0.672i)25-s + (0.863 + 0.503i)27-s + (−0.997 + 0.0669i)29-s + (0.884 − 0.467i)31-s + ⋯
L(s)  = 1  + (−0.643 + 0.765i)3-s + (−0.932 + 0.360i)5-s + (−0.137 − 0.990i)7-s + (−0.170 − 0.985i)9-s + (−0.604 + 0.796i)11-s + (0.935 + 0.352i)13-s + (0.324 − 0.945i)15-s + (−0.723 + 0.690i)17-s + (−0.521 + 0.853i)19-s + (0.846 + 0.532i)21-s + (0.129 + 0.991i)23-s + (0.740 − 0.672i)25-s + (0.863 + 0.503i)27-s + (−0.997 + 0.0669i)29-s + (0.884 − 0.467i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.139 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.139 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6008\)    =    \(2^{3} \cdot 751\)
Sign: $-0.139 - 0.990i$
Analytic conductor: \(27.9010\)
Root analytic conductor: \(27.9010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6008} (989, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6008,\ (0:\ ),\ -0.139 - 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01609624345 + 0.01852311809i\)
\(L(\frac12)\) \(\approx\) \(0.01609624345 + 0.01852311809i\)
\(L(1)\) \(\approx\) \(0.5646461747 + 0.1890968775i\)
\(L(1)\) \(\approx\) \(0.5646461747 + 0.1890968775i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
751 \( 1 \)
good3 \( 1 + (-0.643 + 0.765i)T \)
5 \( 1 + (-0.932 + 0.360i)T \)
7 \( 1 + (-0.137 - 0.990i)T \)
11 \( 1 + (-0.604 + 0.796i)T \)
13 \( 1 + (0.935 + 0.352i)T \)
17 \( 1 + (-0.723 + 0.690i)T \)
19 \( 1 + (-0.521 + 0.853i)T \)
23 \( 1 + (0.129 + 0.991i)T \)
29 \( 1 + (-0.997 + 0.0669i)T \)
31 \( 1 + (0.884 - 0.467i)T \)
37 \( 1 + (0.203 - 0.979i)T \)
41 \( 1 + (-0.992 - 0.125i)T \)
43 \( 1 + (-0.356 - 0.934i)T \)
47 \( 1 + (-0.0544 - 0.998i)T \)
53 \( 1 + (0.425 + 0.904i)T \)
59 \( 1 + (0.301 + 0.953i)T \)
61 \( 1 + (0.570 - 0.821i)T \)
67 \( 1 + (-0.372 - 0.928i)T \)
71 \( 1 + (-0.0878 + 0.996i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (0.521 + 0.853i)T \)
83 \( 1 + (-0.832 + 0.553i)T \)
89 \( 1 + (-0.987 + 0.158i)T \)
97 \( 1 + (0.926 - 0.375i)T \)
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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

−18.09559467229465948066504332604, −17.30963395984166004810834025035, −16.28757032099460427789764184976, −16.17392688509733365225506661918, −15.39382897112397845718963220610, −14.81535070504628617473388754941, −13.60546066124438153146640041694, −13.18510169802945899090844767976, −12.67816635937452602944905651517, −11.896524285814351228314191572015, −11.26310540464462343540147455172, −11.06834713037861281835224244898, −10.04955837838959022055504679726, −8.824818153698187244689055098666, −8.518293724206824588392305878187, −7.98159335851953281410506542722, −7.02619007618567973246422563724, −6.410600227904451784861524641184, −5.780402884203411486686486201594, −4.94752470556209114251674144389, −4.51006845977280153216576282608, −3.17919596545577945304373304725, −2.75407408078827249956820359893, −1.726990924005341199103570283424, −0.71393151161017806716785318771, 0.01038028512188377090667657978, 1.16672031221737150310306261618, 2.20669524079533963174866119512, 3.482439425565005552061950148183, 3.90168623276319322759140452254, 4.28719081363818741497464938536, 5.21572854618309580231255954946, 6.034430414598212423217830323813, 6.80146855536166578137483694469, 7.31878361975972411707917868437, 8.1660597620816107750780197404, 8.8421049864971589750682063173, 9.829185766188539081050731234229, 10.365673194216178683791252903677, 10.88912361624566810178071577178, 11.41924821037764776122731191305, 12.11965639382427054656195376814, 12.90517675384382498066996964308, 13.53924517078059315375600709392, 14.459914910595714000510507441177, 15.185325429778082662013888648488, 15.539594011911384403351537834911, 16.15475640349556066978362005949, 16.974994246401661450994386371758, 17.2374934999299736586715306026

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