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Properties

Label 1-6048-6048.2893-r0-0-0
Degree $1$
Conductor $6048$
Sign $0.411 + 0.911i$
Analytic cond. $28.0867$
Root an. cond. $28.0867$
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  + (0.819 + 0.573i)5-s + (0.573 + 0.819i)11-s + (0.996 + 0.0871i)13-s − 17-s + (0.707 + 0.707i)19-s + (0.642 + 0.766i)23-s + (0.342 + 0.939i)25-s + (−0.0871 − 0.996i)29-s + (−0.939 − 0.342i)31-s + (0.258 + 0.965i)37-s + (0.642 + 0.766i)41-s + (0.422 − 0.906i)43-s + (0.939 − 0.342i)47-s + (−0.258 − 0.965i)53-s + i·55-s + ⋯
L(s)  = 1  + (0.819 + 0.573i)5-s + (0.573 + 0.819i)11-s + (0.996 + 0.0871i)13-s − 17-s + (0.707 + 0.707i)19-s + (0.642 + 0.766i)23-s + (0.342 + 0.939i)25-s + (−0.0871 − 0.996i)29-s + (−0.939 − 0.342i)31-s + (0.258 + 0.965i)37-s + (0.642 + 0.766i)41-s + (0.422 − 0.906i)43-s + (0.939 − 0.342i)47-s + (−0.258 − 0.965i)53-s + i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.411 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.411 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.411 + 0.911i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (2893, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.411 + 0.911i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.140334946 + 1.382379615i\)
\(L(\frac12)\) \(\approx\) \(2.140334946 + 1.382379615i\)
\(L(1)\) \(\approx\) \(1.353084547 + 0.3268361491i\)
\(L(1)\) \(\approx\) \(1.353084547 + 0.3268361491i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.819 + 0.573i)T \)
11 \( 1 + (0.573 + 0.819i)T \)
13 \( 1 + (0.996 + 0.0871i)T \)
17 \( 1 - T \)
19 \( 1 + (0.707 + 0.707i)T \)
23 \( 1 + (0.642 + 0.766i)T \)
29 \( 1 + (-0.0871 - 0.996i)T \)
31 \( 1 + (-0.939 - 0.342i)T \)
37 \( 1 + (0.258 + 0.965i)T \)
41 \( 1 + (0.642 + 0.766i)T \)
43 \( 1 + (0.422 - 0.906i)T \)
47 \( 1 + (0.939 - 0.342i)T \)
53 \( 1 + (-0.258 - 0.965i)T \)
59 \( 1 + (0.0871 - 0.996i)T \)
61 \( 1 + (0.906 + 0.422i)T \)
67 \( 1 + (0.573 - 0.819i)T \)
71 \( 1 + (-0.866 + 0.5i)T \)
73 \( 1 + (0.866 - 0.5i)T \)
79 \( 1 + (-0.173 + 0.984i)T \)
83 \( 1 + (0.0871 + 0.996i)T \)
89 \( 1 - iT \)
97 \( 1 + (-0.939 + 0.342i)T \)
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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

−17.663310778909443312630403044749, −16.923919056098851742889977999173, −16.13455057022322227780775842135, −15.97632015415502764932496863375, −14.87636833847047586962768248750, −14.119451672323537190037130828568, −13.72961732062540444657329150750, −12.92461511532758695163909366839, −12.60092349791507412852158786005, −11.49088015878558244664016773254, −10.95080338080644270084529695196, −10.45572866089604228235950109810, −9.21567736610233611561926230577, −9.0457126571141179472062304780, −8.54495854029431974951878505639, −7.432632900477649222029277548687, −6.70684052974805488147057381014, −5.99597616924758084797243629924, −5.488792042485394780312213688433, −4.63593047832096016880078672295, −3.921054287193996396922490295880, −3.025982740116855760051317366025, −2.26869571723012579141000228, −1.29309352823666740952271971739, −0.72800398099981499221377595042, 1.045457471942046094896780554037, 1.82829034824072926856705669253, 2.41752151671460407271246141858, 3.465348198471370083867754684433, 3.9912276428881519963699828446, 4.97906507672517945752565203883, 5.73157952915820154833415444753, 6.364822347210206596151884796205, 6.9656097707325807326336782755, 7.65282436492106074252303696426, 8.57935165209798792182948916893, 9.37308313953008047194367256930, 9.7213237271515645490209426874, 10.52972631422346527823905461124, 11.29267676604055977519053653524, 11.66696917034261918772173878474, 12.78617224931361443821956960067, 13.23390971044892113666111894545, 13.94037357043886492144451122103, 14.447395753712919878172182690141, 15.30047728827845584120155609803, 15.64371528735548633674761421240, 16.7222364284773725257576526945, 17.17653547045203722853561990750, 17.92266869103797984758271353735

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