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Properties

Label 1-6008-6008.117-r0-0-0
Degree $1$
Conductor $6008$
Sign $0.0147 - 0.999i$
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  + (0.425 + 0.904i)3-s + (0.992 − 0.125i)5-s + (−0.425 − 0.904i)7-s + (−0.637 + 0.770i)9-s + (−0.309 − 0.951i)11-s + (0.929 − 0.368i)13-s + (0.535 + 0.844i)15-s + (−0.187 − 0.982i)17-s + (−0.0627 − 0.998i)19-s + (0.637 − 0.770i)21-s + (0.0627 − 0.998i)23-s + (0.968 − 0.248i)25-s + (−0.968 − 0.248i)27-s + (0.637 − 0.770i)29-s + (−0.637 + 0.770i)31-s + ⋯
L(s)  = 1  + (0.425 + 0.904i)3-s + (0.992 − 0.125i)5-s + (−0.425 − 0.904i)7-s + (−0.637 + 0.770i)9-s + (−0.309 − 0.951i)11-s + (0.929 − 0.368i)13-s + (0.535 + 0.844i)15-s + (−0.187 − 0.982i)17-s + (−0.0627 − 0.998i)19-s + (0.637 − 0.770i)21-s + (0.0627 − 0.998i)23-s + (0.968 − 0.248i)25-s + (−0.968 − 0.248i)27-s + (0.637 − 0.770i)29-s + (−0.637 + 0.770i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0147 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0147 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6008\)    =    \(2^{3} \cdot 751\)
Sign: $0.0147 - 0.999i$
Analytic conductor: \(27.9010\)
Root analytic conductor: \(27.9010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6008} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6008,\ (0:\ ),\ 0.0147 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.338392892 - 1.318768139i\)
\(L(\frac12)\) \(\approx\) \(1.338392892 - 1.318768139i\)
\(L(1)\) \(\approx\) \(1.282270474 - 0.09841357722i\)
\(L(1)\) \(\approx\) \(1.282270474 - 0.09841357722i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
751 \( 1 \)
good3 \( 1 + (0.425 + 0.904i)T \)
5 \( 1 + (0.992 - 0.125i)T \)
7 \( 1 + (-0.425 - 0.904i)T \)
11 \( 1 + (-0.309 - 0.951i)T \)
13 \( 1 + (0.929 - 0.368i)T \)
17 \( 1 + (-0.187 - 0.982i)T \)
19 \( 1 + (-0.0627 - 0.998i)T \)
23 \( 1 + (0.0627 - 0.998i)T \)
29 \( 1 + (0.637 - 0.770i)T \)
31 \( 1 + (-0.637 + 0.770i)T \)
37 \( 1 + (-0.968 + 0.248i)T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 + (0.992 + 0.125i)T \)
47 \( 1 + (-0.992 + 0.125i)T \)
53 \( 1 + (-0.309 + 0.951i)T \)
59 \( 1 + (-0.876 - 0.481i)T \)
61 \( 1 + (-0.309 + 0.951i)T \)
67 \( 1 + (0.637 - 0.770i)T \)
71 \( 1 + (0.535 - 0.844i)T \)
73 \( 1 + T \)
79 \( 1 + (0.0627 - 0.998i)T \)
83 \( 1 + (0.809 - 0.587i)T \)
89 \( 1 + (-0.992 + 0.125i)T \)
97 \( 1 + (-0.425 + 0.904i)T \)
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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

−17.93103310468761499213648378522, −17.48176445033066449644124116876, −16.71408136881330287885255895328, −15.765741381870216260899465909790, −15.20181298340256468013011359468, −14.43953114004562715988948101774, −13.96656447491261389522890397432, −13.17521590894463210515876652085, −12.62948343319067466029310211654, −12.32824685809181022411952194302, −11.3148077217323969042941224934, −10.56785198134807485137836241204, −9.67786144567668707243919611557, −9.24643755206355729268615928156, −8.51890896847623842687505545991, −7.92088887471833840199533409689, −6.90007995092031392428270203931, −6.50739550524355388829499645906, −5.69411124352102662981181068495, −5.36401501606733171716822438656, −3.949664051290285156388836560439, −3.31187944961911588843464614143, −2.35223360395459246377652185787, −1.85934364383469495705742121417, −1.33765332238027499337235284288, 0.407382152873193562345047401212, 1.30897157609152966828947308297, 2.57980139431137644302760993948, 2.980100846180208281001549586678, 3.7301268168542024559957613825, 4.70264848077577665592273047826, 5.08906085798322208224928525598, 6.129852451469022300203893468496, 6.51849120130833121378085978701, 7.58989835783188395066310609947, 8.382279117676405490135277827993, 9.02387520997579399266915741055, 9.51339748335757355782279478408, 10.30792750467175245262909920511, 10.83621629869824873152884156994, 11.16904740636018092719583008131, 12.45741823197067984423723919917, 13.29442984844961722578865795727, 13.764709395536231426116250058115, 13.99172108303329575356056289330, 14.91650339130936876401931328232, 15.867358677348519143637499038469, 16.13295885330462735644913985466, 16.76472579252836176102229245632, 17.45615066579882795345635438762

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