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Properties

Label 1-6048-6048.5891-r0-0-0
Degree $1$
Conductor $6048$
Sign $0.879 + 0.476i$
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 + (0.573 − 0.819i)13-s + (−0.5 + 0.866i)17-s + (0.258 − 0.965i)19-s + (0.642 + 0.766i)23-s + (0.342 + 0.939i)25-s + (−0.819 + 0.573i)29-s + (−0.173 − 0.984i)31-s + (−0.707 + 0.707i)37-s + (0.984 − 0.173i)41-s + (−0.996 + 0.0871i)43-s + (−0.173 + 0.984i)47-s + (0.965 + 0.258i)53-s − i·55-s + ⋯
L(s)  = 1  + (0.819 + 0.573i)5-s + (−0.573 − 0.819i)11-s + (0.573 − 0.819i)13-s + (−0.5 + 0.866i)17-s + (0.258 − 0.965i)19-s + (0.642 + 0.766i)23-s + (0.342 + 0.939i)25-s + (−0.819 + 0.573i)29-s + (−0.173 − 0.984i)31-s + (−0.707 + 0.707i)37-s + (0.984 − 0.173i)41-s + (−0.996 + 0.0871i)43-s + (−0.173 + 0.984i)47-s + (0.965 + 0.258i)53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.879 + 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.879 + 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.879 + 0.476i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (5891, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.879 + 0.476i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.976800787 + 0.5011657934i\)
\(L(\frac12)\) \(\approx\) \(1.976800787 + 0.5011657934i\)
\(L(1)\) \(\approx\) \(1.217668150 + 0.09377722639i\)
\(L(1)\) \(\approx\) \(1.217668150 + 0.09377722639i\)

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 \( 1 + (0.573 - 0.819i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.258 - 0.965i)T \)
23 \( 1 + (0.642 + 0.766i)T \)
29 \( 1 + (-0.819 + 0.573i)T \)
31 \( 1 + (-0.173 - 0.984i)T \)
37 \( 1 + (-0.707 + 0.707i)T \)
41 \( 1 + (0.984 - 0.173i)T \)
43 \( 1 + (-0.996 + 0.0871i)T \)
47 \( 1 + (-0.173 + 0.984i)T \)
53 \( 1 + (0.965 + 0.258i)T \)
59 \( 1 + (0.906 - 0.422i)T \)
61 \( 1 + (0.819 - 0.573i)T \)
67 \( 1 + (-0.996 - 0.0871i)T \)
71 \( 1 + (-0.866 + 0.5i)T \)
73 \( 1 + iT \)
79 \( 1 + (0.766 + 0.642i)T \)
83 \( 1 + (-0.819 + 0.573i)T \)
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

−17.91371751953040468612892399940, −16.88107542442194701164499887782, −16.35358004178114903049180217196, −15.92862533906984668192838841835, −14.88078131004620004171943725933, −14.4317594099076514575747592914, −13.42190975427682174118942582346, −13.33953953995925987323955701300, −12.35398431135542990575275811226, −11.88797534808809906001445488670, −10.9652871582519800490245505157, −10.24191197749918435771001489755, −9.71858252315038704747705100239, −8.898286455404934817704471228158, −8.57792006264526076475236277538, −7.44341428820428911560216899200, −6.93161829601827202867872913810, −6.08927650662413399685849678307, −5.38960665198635591483939906689, −4.76327373234517625539975447151, −4.0937635252863040072947955279, −3.09808619762157588044879786103, −2.10038598049172430752592494430, −1.754058166573901423448749153005, −0.6257703153586968540577874481, 0.80112976511149292841351593272, 1.70336822850193408294144665865, 2.58914745567033246662848316714, 3.19121350724283410811687764249, 3.87714211757797030811178083427, 5.077445194769764346874722966858, 5.588371557706537291537731781975, 6.18332260909465859020779136199, 6.94170921944605872049084596448, 7.664687261907424978710798873101, 8.48464711229188570602456850327, 9.081783953633758690349005344977, 9.84680958080178841019818964615, 10.58432874486105428924578122473, 11.053147591852458199034513112, 11.55228748856661410802786831993, 12.92464277227640672612623089430, 13.13167811140073495024200830957, 13.65534634164701500537953041722, 14.52544516394417907935433534481, 15.1694412537116695069282245971, 15.67411572433023361028131387938, 16.46662364205997285318025293842, 17.37558147865785751207876839761, 17.59009570143943782835505020743

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