Properties

Label 1-6048-6048.677-r0-0-0
Degree $1$
Conductor $6048$
Sign $0.986 - 0.162i$
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  + (−0.906 − 0.422i)5-s + (−0.422 − 0.906i)11-s + (0.422 − 0.906i)13-s + (0.5 + 0.866i)17-s + (−0.258 − 0.965i)19-s + (0.984 + 0.173i)23-s + (0.642 + 0.766i)25-s + (−0.906 + 0.422i)29-s + (0.939 − 0.342i)31-s + (0.707 + 0.707i)37-s + (0.342 + 0.939i)41-s + (0.573 − 0.819i)43-s + (0.939 + 0.342i)47-s + (−0.965 + 0.258i)53-s i·55-s + ⋯
L(s)  = 1  + (−0.906 − 0.422i)5-s + (−0.422 − 0.906i)11-s + (0.422 − 0.906i)13-s + (0.5 + 0.866i)17-s + (−0.258 − 0.965i)19-s + (0.984 + 0.173i)23-s + (0.642 + 0.766i)25-s + (−0.906 + 0.422i)29-s + (0.939 − 0.342i)31-s + (0.707 + 0.707i)37-s + (0.342 + 0.939i)41-s + (0.573 − 0.819i)43-s + (0.939 + 0.342i)47-s + (−0.965 + 0.258i)53-s i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.986 - 0.162i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (677, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.986 - 0.162i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.413324416 - 0.1155209185i\)
\(L(\frac12)\) \(\approx\) \(1.413324416 - 0.1155209185i\)
\(L(1)\) \(\approx\) \(0.9390986285 - 0.1132176409i\)
\(L(1)\) \(\approx\) \(0.9390986285 - 0.1132176409i\)

Euler product

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

−17.85069963089017573212072041310, −16.889553941410408136977236909821, −16.45358646423540737950821408051, −15.56668786574094856140013072587, −15.32256716061529193668484512257, −14.264486438583800109199617094606, −14.141704174903582525533145525, −12.94995460831412656698838503266, −12.41644060883297217101574422817, −11.80495314481990218806896640868, −11.088959132806815593495215874498, −10.60969414890438199722789890727, −9.647552754294227069300606824855, −9.192193443664536976712427014628, −8.15680542748759190753382510937, −7.673487500426439135524544761010, −7.024539195222661942132306458852, −6.40061043970075164209929044302, −5.45715156890318005808696185264, −4.60590904471349894776271088659, −4.08207728166771948787138314003, −3.284346271575604487264673178793, −2.50575522117064477729122299524, −1.67341716967227465060465183309, −0.55476731451262536495857366092, 0.72485085226880735857925732375, 1.242910965476137468108454681, 2.69817544440348309861165208301, 3.16029343503296545894570696477, 3.985476517196228507557406068904, 4.6774930530756998675319945743, 5.52804518156176005962288052579, 6.035660947777572343181344330730, 7.071383614867885481150168797123, 7.72747040746910071222992967044, 8.36801674552407129033651801560, 8.78979932332353175551505635217, 9.67386985341723653713313221477, 10.655454876371091406959931215076, 11.0483426865502310164259372661, 11.6421762803919322870541894723, 12.6000135529721224874578327366, 12.997465212498732793581916245857, 13.57394292855040292457464659058, 14.5530066594798300185924628579, 15.345053854950626563646304345676, 15.52344044708107241456059546203, 16.38904239746922636820769837867, 16.99958880761418935037634423718, 17.518475616166117370798791666136

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