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Properties

Label 1-6048-6048.205-r0-0-0
Degree $1$
Conductor $6048$
Sign $-0.776 - 0.629i$
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  + (−0.573 − 0.819i)5-s + (−0.819 − 0.573i)11-s + (0.819 − 0.573i)13-s + (0.5 + 0.866i)17-s + (−0.965 + 0.258i)19-s + (−0.642 + 0.766i)23-s + (−0.342 + 0.939i)25-s + (0.573 − 0.819i)29-s + (0.173 − 0.984i)31-s + (0.707 − 0.707i)37-s + (0.984 + 0.173i)41-s + (−0.0871 + 0.996i)43-s + (−0.173 − 0.984i)47-s + (−0.258 − 0.965i)53-s + i·55-s + ⋯
L(s)  = 1  + (−0.573 − 0.819i)5-s + (−0.819 − 0.573i)11-s + (0.819 − 0.573i)13-s + (0.5 + 0.866i)17-s + (−0.965 + 0.258i)19-s + (−0.642 + 0.766i)23-s + (−0.342 + 0.939i)25-s + (0.573 − 0.819i)29-s + (0.173 − 0.984i)31-s + (0.707 − 0.707i)37-s + (0.984 + 0.173i)41-s + (−0.0871 + 0.996i)43-s + (−0.173 − 0.984i)47-s + (−0.258 − 0.965i)53-s + i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.776 - 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.776 - 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $-0.776 - 0.629i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ -0.776 - 0.629i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3079584642 - 0.8684964138i\)
\(L(\frac12)\) \(\approx\) \(0.3079584642 - 0.8684964138i\)
\(L(1)\) \(\approx\) \(0.8389235754 - 0.2335015550i\)
\(L(1)\) \(\approx\) \(0.8389235754 - 0.2335015550i\)

Euler product

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

−18.00607080695760081922550906259, −17.50659603407394962360709598233, −16.39245109397025885700435374964, −15.98111890621604192160165690515, −15.44933435687185890704351796364, −14.57316333256050282276853028871, −14.21381040338922559493969763468, −13.42413839584007541329522252456, −12.598129219625633675995671017376, −12.04018812897797525535428605219, −11.31772836923347514286230844610, −10.56775916835016813341619085006, −10.302991254998899355720979369101, −9.27942665688043088634362794468, −8.54543974901309038802378952638, −7.86329439647379579149172570232, −7.20667876661892050309600263830, −6.55857219451057516441656312074, −5.954586096918308650189602655, −4.82776717308563903013398820826, −4.378202582463431185461392220440, −3.45806995159999842118033913652, −2.75611049788874902567719031636, −2.11772730786745610359871995385, −0.9743316842409395957229472063, 0.28049134140355227507562855348, 1.12418289920083205343042865353, 2.052440253273966627988636615343, 3.00099037590201580717884804918, 3.90536782486852240429201493824, 4.23234166547889168109197885747, 5.374484251544494091373226656438, 5.79285518959114246064238695077, 6.5004258738464527159163842289, 7.81665330644258868957497844618, 8.01936080590370856250777083989, 8.51607247850667345881169140556, 9.483180263131959254156683592149, 10.138767772358352004204785444399, 10.96598433884667393636250841592, 11.41752357370881216410839521712, 12.279509981197336104914936331927, 13.01152584455925521025593489928, 13.19735632065101115244514932889, 14.15231009126561628645423904639, 14.99839374859900448593175911790, 15.600014743849052894917375777003, 16.064887404664590277808438963248, 16.76335990583105795550428453169, 17.328558941125742178533118359532

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