Properties

Label 1-6027-6027.5396-r0-0-0
Degree $1$
Conductor $6027$
Sign $0.514 - 0.857i$
Analytic cond. $27.9892$
Root an. cond. $27.9892$
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  + (0.393 − 0.919i)2-s + (−0.691 − 0.722i)4-s + (−0.550 + 0.834i)5-s + (−0.936 + 0.351i)8-s + (0.550 + 0.834i)10-s + (−0.995 − 0.0896i)11-s + (−0.393 + 0.919i)13-s + (−0.0448 + 0.998i)16-s + (0.691 − 0.722i)17-s + (−0.809 + 0.587i)19-s + (0.983 − 0.178i)20-s + (−0.473 + 0.880i)22-s + (−0.983 − 0.178i)23-s + (−0.393 − 0.919i)25-s + (0.691 + 0.722i)26-s + ⋯
L(s)  = 1  + (0.393 − 0.919i)2-s + (−0.691 − 0.722i)4-s + (−0.550 + 0.834i)5-s + (−0.936 + 0.351i)8-s + (0.550 + 0.834i)10-s + (−0.995 − 0.0896i)11-s + (−0.393 + 0.919i)13-s + (−0.0448 + 0.998i)16-s + (0.691 − 0.722i)17-s + (−0.809 + 0.587i)19-s + (0.983 − 0.178i)20-s + (−0.473 + 0.880i)22-s + (−0.983 − 0.178i)23-s + (−0.393 − 0.919i)25-s + (0.691 + 0.722i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.514 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.514 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $0.514 - 0.857i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (5396, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ 0.514 - 0.857i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7220607473 - 0.4087028646i\)
\(L(\frac12)\) \(\approx\) \(0.7220607473 - 0.4087028646i\)
\(L(1)\) \(\approx\) \(0.7647210201 - 0.2887582291i\)
\(L(1)\) \(\approx\) \(0.7647210201 - 0.2887582291i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.393 - 0.919i)T \)
5 \( 1 + (-0.550 + 0.834i)T \)
11 \( 1 + (-0.995 - 0.0896i)T \)
13 \( 1 + (-0.393 + 0.919i)T \)
17 \( 1 + (0.691 - 0.722i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (-0.983 - 0.178i)T \)
29 \( 1 + (0.134 - 0.990i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + (0.134 - 0.990i)T \)
43 \( 1 + (-0.963 + 0.266i)T \)
47 \( 1 + (0.393 - 0.919i)T \)
53 \( 1 + (-0.691 - 0.722i)T \)
59 \( 1 + (-0.963 + 0.266i)T \)
61 \( 1 + (-0.983 + 0.178i)T \)
67 \( 1 + (-0.309 - 0.951i)T \)
71 \( 1 + (0.134 + 0.990i)T \)
73 \( 1 + (0.222 - 0.974i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.222 + 0.974i)T \)
89 \( 1 + (0.393 + 0.919i)T \)
97 \( 1 + (0.309 + 0.951i)T \)
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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

−17.54026052880859094783697445452, −17.03417302413448819222336871978, −16.5000014587300633879326572120, −15.64944170726375392233849808630, −15.39618474530641992187062259907, −14.74472438306994846151717900325, −13.90289570320521804462700746623, −13.08662634455611999763652835172, −12.73504327680880930543607064829, −12.24194745638287906344329465369, −11.36647341753984363232844813647, −10.408958005170271260416500382535, −9.77027299276294355825974296290, −8.84270339416979046615880866386, −8.27643091339796461009049924672, −7.74319893961014726633573447219, −7.27998332294865608711663778280, −6.09079094556975381142774236887, −5.69022377784039769981380585954, −4.79717894044566343639098299163, −4.48263963325159676420801934759, −3.46857256583778149173137124243, −2.88543097844759494297544194377, −1.67445865350607819807657832827, −0.43165842420745806013078484481, 0.38601966259615373945940076575, 1.78292220014732068280507452431, 2.347295539252383281078048309645, 3.08557232588213450887228960048, 3.78217120802638893307852439846, 4.45220115425257055902757858578, 5.19246465300010837321782292614, 6.035624117610957387671533989631, 6.697301586637811012864881494698, 7.65590811668605789689690577218, 8.18022925025504457466557758539, 9.13130180894747196583992079737, 9.91055241539041490577668852785, 10.43433242518960561455555667905, 10.95266834361876388948348487350, 11.8730131576931278084444131133, 12.04884125950060562241639005700, 12.91320036179171611047074620803, 13.74807232430395550584192911723, 14.25130547358576707301461126419, 14.75399153245217006923512268586, 15.537541726939925890455586181024, 16.14754555135009463362095652045, 16.981564706817431256971542229505, 17.99546098625072542262579645381

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