Properties

Label 1-6048-6048.5155-r0-0-0
Degree $1$
Conductor $6048$
Sign $0.677 - 0.735i$
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  + (0.996 − 0.0871i)5-s + (−0.0871 + 0.996i)11-s + (−0.906 − 0.422i)13-s + 17-s + (−0.707 − 0.707i)19-s + (0.342 + 0.939i)23-s + (0.984 − 0.173i)25-s + (−0.422 − 0.906i)29-s + (0.173 − 0.984i)31-s + (0.965 + 0.258i)37-s + (0.342 + 0.939i)41-s + (−0.819 − 0.573i)43-s + (−0.173 − 0.984i)47-s + (−0.965 − 0.258i)53-s i·55-s + ⋯
L(s)  = 1  + (0.996 − 0.0871i)5-s + (−0.0871 + 0.996i)11-s + (−0.906 − 0.422i)13-s + 17-s + (−0.707 − 0.707i)19-s + (0.342 + 0.939i)23-s + (0.984 − 0.173i)25-s + (−0.422 − 0.906i)29-s + (0.173 − 0.984i)31-s + (0.965 + 0.258i)37-s + (0.342 + 0.939i)41-s + (−0.819 − 0.573i)43-s + (−0.173 − 0.984i)47-s + (−0.965 − 0.258i)53-s i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.677 - 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.677 - 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.677 - 0.735i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (5155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.677 - 0.735i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.829732979 - 0.8016939183i\)
\(L(\frac12)\) \(\approx\) \(1.829732979 - 0.8016939183i\)
\(L(1)\) \(\approx\) \(1.225953410 - 0.09466177552i\)
\(L(1)\) \(\approx\) \(1.225953410 - 0.09466177552i\)

Euler product

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

−17.72912373101237913887985627563, −17.08926884916887491864370063918, −16.42771665728221201315750184970, −16.210114666857081339909943734150, −14.77478287271989295985911837409, −14.558895014670150194070937928005, −14.01080905021886195325578908218, −13.154650690474058664595513780340, −12.59722740314499949914031077810, −11.98134241242065784244142502924, −11.00193952397311706219606327657, −10.48369217495908202906048690853, −9.8803069063988222327295108912, −9.13131675550137225284752210798, −8.5599238218268850150268951606, −7.7473931796867431874568340456, −6.91672605049392376311524748471, −6.25581580505603132080462053356, −5.61796836116474690940287971662, −4.99997445529511504062207510623, −4.13487517616051388513942567251, −3.129796766959064840098055894149, −2.61074867629957728468373843118, −1.69007688357513444577315526411, −0.93535513069501612785947731883, 0.538741632029062911815485047310, 1.66003112237226490419740971568, 2.25609687097758215792292840563, 2.94536990259703453094720786367, 3.94296851901961333281758815070, 4.933814801206050087654865987294, 5.21242782529886906731705767466, 6.16084229680211569168815672966, 6.75608339430078406715255208616, 7.630115625327971980178662599421, 8.096290597701475839536917015144, 9.244646533275872046099667657577, 9.75242974442110821289880399534, 10.01513094658337863368941977497, 10.9857527421001264804678727049, 11.70917442905010947388228861460, 12.51569175135968605790628813478, 13.03588211340026580965415879675, 13.56376150368558118774163185992, 14.43248564120726148372562463025, 15.05681242674065109296297490036, 15.37238101066062148338145543899, 16.64045699595193277451469144932, 16.96824212762019338560930682715, 17.5848928199711945385835097568

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