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Properties

Label 1-6012-6012.43-r0-0-0
Degree $1$
Conductor $6012$
Sign $0.564 - 0.825i$
Analytic cond. $27.9196$
Root an. cond. $27.9196$
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  + (0.624 − 0.780i)5-s + (0.998 + 0.0630i)7-s + (0.543 − 0.839i)11-s + (−0.398 + 0.917i)13-s + (0.169 − 0.985i)17-s + (0.800 − 0.599i)19-s + (−0.194 − 0.980i)23-s + (−0.219 − 0.975i)25-s + (−0.194 + 0.980i)29-s + (0.999 + 0.0126i)31-s + (0.672 − 0.739i)35-s + (0.982 − 0.188i)37-s + (0.477 + 0.878i)41-s + (0.0819 − 0.996i)43-s + (−0.574 + 0.818i)47-s + ⋯
L(s)  = 1  + (0.624 − 0.780i)5-s + (0.998 + 0.0630i)7-s + (0.543 − 0.839i)11-s + (−0.398 + 0.917i)13-s + (0.169 − 0.985i)17-s + (0.800 − 0.599i)19-s + (−0.194 − 0.980i)23-s + (−0.219 − 0.975i)25-s + (−0.194 + 0.980i)29-s + (0.999 + 0.0126i)31-s + (0.672 − 0.739i)35-s + (0.982 − 0.188i)37-s + (0.477 + 0.878i)41-s + (0.0819 − 0.996i)43-s + (−0.574 + 0.818i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.564 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.564 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6012\)    =    \(2^{2} \cdot 3^{2} \cdot 167\)
Sign: $0.564 - 0.825i$
Analytic conductor: \(27.9196\)
Root analytic conductor: \(27.9196\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6012} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6012,\ (0:\ ),\ 0.564 - 0.825i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.468103061 - 1.301912275i\)
\(L(\frac12)\) \(\approx\) \(2.468103061 - 1.301912275i\)
\(L(1)\) \(\approx\) \(1.437593769 - 0.3288022342i\)
\(L(1)\) \(\approx\) \(1.437593769 - 0.3288022342i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
167 \( 1 \)
good5 \( 1 + (0.624 - 0.780i)T \)
7 \( 1 + (0.998 + 0.0630i)T \)
11 \( 1 + (0.543 - 0.839i)T \)
13 \( 1 + (-0.398 + 0.917i)T \)
17 \( 1 + (0.169 - 0.985i)T \)
19 \( 1 + (0.800 - 0.599i)T \)
23 \( 1 + (-0.194 - 0.980i)T \)
29 \( 1 + (-0.194 + 0.980i)T \)
31 \( 1 + (0.999 + 0.0126i)T \)
37 \( 1 + (0.982 - 0.188i)T \)
41 \( 1 + (0.477 + 0.878i)T \)
43 \( 1 + (0.0819 - 0.996i)T \)
47 \( 1 + (-0.574 + 0.818i)T \)
53 \( 1 + (0.700 + 0.713i)T \)
59 \( 1 + (0.938 + 0.345i)T \)
61 \( 1 + (-0.0441 + 0.999i)T \)
67 \( 1 + (-0.363 + 0.931i)T \)
71 \( 1 + (0.862 + 0.505i)T \)
73 \( 1 + (-0.614 - 0.788i)T \)
79 \( 1 + (0.107 + 0.994i)T \)
83 \( 1 + (-0.292 - 0.956i)T \)
89 \( 1 + (-0.243 - 0.969i)T \)
97 \( 1 + (-0.999 + 0.0126i)T \)
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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

−17.86526797023718032248342399509, −17.29033272015800582450411690896, −16.79888616507730253320621766938, −15.5730852106208952019802712786, −15.08165104919410085315876069466, −14.6000326963418990927037621288, −14.00532297125968825090954730089, −13.31314926902219001248064088725, −12.546831708239887059485027517672, −11.7512807481212504388522225747, −11.27951979336589719930263837037, −10.39261463455730122141969113905, −9.91176517090792117462625594740, −9.411887273933092020953944785093, −8.19675988342439119090881013964, −7.812550553453205134805834858122, −7.12471843870239746589731674375, −6.25945957285628088042252276300, −5.63332970421764620609774225143, −4.98258391242193125845177075824, −4.05650995371174378277548598877, −3.38485278881547525024779420364, −2.3879598216447967099104520507, −1.81815858460231798050723453066, −1.03381779266043971403964645039, 0.83037090128434616054555733609, 1.28247425960535224296658089944, 2.30998473677266277736874545327, 2.902944393382928017163045118860, 4.200676184322593160111174682789, 4.6291420578373798636639537755, 5.33017896084554591665430893229, 5.987853172159164861213290155455, 6.84291367941612925349162702107, 7.546320477674949423573195369371, 8.46258260789273697914349198185, 8.906198641962646200235087529300, 9.48785468012082891725532253316, 10.24570018992288216687542428574, 11.19245222695404445653763122945, 11.69183814865582307365399954036, 12.18642496812141431079102772208, 13.138705511931291349035325772746, 13.798662595400417615312652659340, 14.25823358983500799492183749892, 14.74635362059696923446636262594, 15.90352984710042043577534028231, 16.38526517817637592901331100034, 16.8892801742935699196100568260, 17.60234505884625309071547715010

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