Properties

Label 1-6008-6008.1077-r0-0-0
Degree $1$
Conductor $6008$
Sign $-0.942 - 0.332i$
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  + (−0.813 + 0.580i)3-s + (0.734 − 0.678i)5-s + (0.112 − 0.993i)7-s + (0.324 − 0.945i)9-s + (0.957 − 0.289i)11-s + (−0.717 + 0.696i)13-s + (−0.203 + 0.979i)15-s + (−0.970 − 0.240i)17-s + (0.935 + 0.352i)19-s + (0.485 + 0.874i)21-s + (−0.998 − 0.0586i)23-s + (0.0795 − 0.996i)25-s + (0.285 + 0.958i)27-s + (−0.906 − 0.421i)29-s + (0.999 + 0.0167i)31-s + ⋯
L(s)  = 1  + (−0.813 + 0.580i)3-s + (0.734 − 0.678i)5-s + (0.112 − 0.993i)7-s + (0.324 − 0.945i)9-s + (0.957 − 0.289i)11-s + (−0.717 + 0.696i)13-s + (−0.203 + 0.979i)15-s + (−0.970 − 0.240i)17-s + (0.935 + 0.352i)19-s + (0.485 + 0.874i)21-s + (−0.998 − 0.0586i)23-s + (0.0795 − 0.996i)25-s + (0.285 + 0.958i)27-s + (−0.906 − 0.421i)29-s + (0.999 + 0.0167i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.942 - 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.942 - 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6008\)    =    \(2^{3} \cdot 751\)
Sign: $-0.942 - 0.332i$
Analytic conductor: \(27.9010\)
Root analytic conductor: \(27.9010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6008} (1077, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6008,\ (0:\ ),\ -0.942 - 0.332i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1002100667 - 0.5849862356i\)
\(L(\frac12)\) \(\approx\) \(0.1002100667 - 0.5849862356i\)
\(L(1)\) \(\approx\) \(0.8216560195 - 0.1317065752i\)
\(L(1)\) \(\approx\) \(0.8216560195 - 0.1317065752i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
751 \( 1 \)
good3 \( 1 + (-0.813 + 0.580i)T \)
5 \( 1 + (0.734 - 0.678i)T \)
7 \( 1 + (0.112 - 0.993i)T \)
11 \( 1 + (0.957 - 0.289i)T \)
13 \( 1 + (-0.717 + 0.696i)T \)
17 \( 1 + (-0.970 - 0.240i)T \)
19 \( 1 + (0.935 + 0.352i)T \)
23 \( 1 + (-0.998 - 0.0586i)T \)
29 \( 1 + (-0.906 - 0.421i)T \)
31 \( 1 + (0.999 + 0.0167i)T \)
37 \( 1 + (-0.521 - 0.853i)T \)
41 \( 1 + (0.728 + 0.684i)T \)
43 \( 1 + (-0.0125 + 0.999i)T \)
47 \( 1 + (0.418 + 0.908i)T \)
53 \( 1 + (-0.876 - 0.481i)T \)
59 \( 1 + (-0.932 + 0.360i)T \)
61 \( 1 + (0.0209 + 0.999i)T \)
67 \( 1 + (0.121 - 0.992i)T \)
71 \( 1 + (0.212 - 0.977i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (-0.935 + 0.352i)T \)
83 \( 1 + (-0.783 - 0.621i)T \)
89 \( 1 + (0.859 - 0.510i)T \)
97 \( 1 + (0.804 - 0.594i)T \)
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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

−17.88905566371426681313744687760, −17.34619439152167012261915282940, −17.17048291653300582789214432294, −15.92879673125888395486464782801, −15.465731930282627691689335548337, −14.70637591086844378199212021545, −14.00632676129857426410014749844, −13.41811051210137709486335165131, −12.61358139464107910830990046033, −12.03808765415216419921084079364, −11.5358148496126836478241125669, −10.80587327233791024837854998804, −10.08182777342341126347318702361, −9.46641750116859346919697707173, −8.7220541386656245421586275860, −7.78017222180288294950026106240, −7.07107541620471448246853056620, −6.50179259289988816874357794818, −5.85200431092178077695812766542, −5.32063210237941737991665038498, −4.59177832695612062346791819811, −3.463584435493505514164893843891, −2.48016455622509924448378301029, −2.03488621156435753122242055781, −1.22764917296639144015621778804, 0.16862415309240570248861570384, 1.175691077250741302000850909041, 1.76087656379953286181985121721, 2.967246705081403772704012974095, 4.17948788209087794961540389716, 4.27283777191114620112467196870, 5.073598402264354896016536523394, 5.97865984584012413261819653884, 6.40084689487720647488839152330, 7.19509134228436962245864467877, 8.01087087257437977121856304257, 9.15988404093680127383155178779, 9.43716942572429119362671668621, 10.03191257833702988781883952899, 10.80132975203566707121049861339, 11.52977738984144788651833095509, 11.98116656591246266145746237171, 12.77939173288075321341336720728, 13.54961002599155421194662695704, 14.19325582108805304669601399887, 14.60388184461253653626471228673, 15.844570753674511433241349213888, 16.18630628178845319773011853122, 16.86037825905299097740002279013, 17.32309586529899192799178282676

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