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Properties

Label 1-6048-6048.1213-r0-0-0
Degree $1$
Conductor $6048$
Sign $0.987 + 0.157i$
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  + (0.422 − 0.906i)5-s + (−0.906 + 0.422i)11-s + (0.906 + 0.422i)13-s + (0.5 + 0.866i)17-s + (0.965 − 0.258i)19-s + (0.984 + 0.173i)23-s + (−0.642 − 0.766i)25-s + (−0.422 − 0.906i)29-s + (−0.939 + 0.342i)31-s + (−0.707 + 0.707i)37-s + (−0.342 − 0.939i)41-s + (0.819 + 0.573i)43-s + (0.939 + 0.342i)47-s + (0.258 + 0.965i)53-s + i·55-s + ⋯
L(s)  = 1  + (0.422 − 0.906i)5-s + (−0.906 + 0.422i)11-s + (0.906 + 0.422i)13-s + (0.5 + 0.866i)17-s + (0.965 − 0.258i)19-s + (0.984 + 0.173i)23-s + (−0.642 − 0.766i)25-s + (−0.422 − 0.906i)29-s + (−0.939 + 0.342i)31-s + (−0.707 + 0.707i)37-s + (−0.342 − 0.939i)41-s + (0.819 + 0.573i)43-s + (0.939 + 0.342i)47-s + (0.258 + 0.965i)53-s + i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.987 + 0.157i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (1213, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.987 + 0.157i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.009446714 + 0.1589902822i\)
\(L(\frac12)\) \(\approx\) \(2.009446714 + 0.1589902822i\)
\(L(1)\) \(\approx\) \(1.197507981 - 0.06135584278i\)
\(L(1)\) \(\approx\) \(1.197507981 - 0.06135584278i\)

Euler product

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

−17.94089636635037056810047796805, −17.02859717920159065847559478494, −16.27573061630492071687268532949, −15.77522164884540881917999661749, −15.04393933460924237545324314893, −14.37499491519679855918293493620, −13.73020026609938227501180895429, −13.26448297233337797038209685000, −12.50061021402746211380911725416, −11.60653235200576015857034809614, −10.85230136110721277267344790062, −10.647475802151000398415813614621, −9.729769137494948256197336778409, −9.116578354055215313725363195202, −8.30831678490467577512180366189, −7.358760785465764386038437408603, −7.1754858435094468405402077428, −6.05053122074244574841341500804, −5.53468171451543096184376687064, −4.99342251836042479779149222341, −3.63041887331673828231369390148, −3.24386170858361802784149225106, −2.55407220179393343233381726456, −1.61370571000135274902944676801, −0.62651424049074095023399646745, 0.85051207233899992336854789612, 1.54900703025066896198481602331, 2.31934622048230422285909719583, 3.31445358004142382462220333635, 4.05015429031567187549601876231, 4.90387290196318738142097859685, 5.47507354937031745614395396070, 6.046707121065556174478149736478, 7.00305794447100566647909806586, 7.77248953974116842009312295207, 8.35329319692773131901363952941, 9.210048037680314095437862912338, 9.523181533237247498073205530, 10.5408202569481126489803057070, 10.955013738887865525716752520053, 11.96714451637610568877465930952, 12.50365284273715430893658933380, 13.16255101205101102865023884400, 13.665922892911697778046666208778, 14.30390615572155096947858660419, 15.38479224117350560681714573504, 15.6773666437538553202389576309, 16.43708445844889133796569025183, 17.16266327067136984879593742993, 17.5054844545556082126567092051

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