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Properties

Label 1-6048-6048.643-r0-0-0
Degree $1$
Conductor $6048$
Sign $-0.847 - 0.531i$
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  + (0.0871 − 0.996i)5-s + (−0.996 + 0.0871i)11-s + (−0.573 + 0.819i)13-s + (−0.5 − 0.866i)17-s + (0.965 − 0.258i)19-s + (−0.342 + 0.939i)23-s + (−0.984 − 0.173i)25-s + (0.819 − 0.573i)29-s + (−0.939 − 0.342i)31-s + (0.965 + 0.258i)37-s + (0.984 − 0.173i)41-s + (−0.996 + 0.0871i)43-s + (0.939 − 0.342i)47-s + (0.707 − 0.707i)53-s − i·55-s + ⋯
L(s)  = 1  + (0.0871 − 0.996i)5-s + (−0.996 + 0.0871i)11-s + (−0.573 + 0.819i)13-s + (−0.5 − 0.866i)17-s + (0.965 − 0.258i)19-s + (−0.342 + 0.939i)23-s + (−0.984 − 0.173i)25-s + (0.819 − 0.573i)29-s + (−0.939 − 0.342i)31-s + (0.965 + 0.258i)37-s + (0.984 − 0.173i)41-s + (−0.996 + 0.0871i)43-s + (0.939 − 0.342i)47-s + (0.707 − 0.707i)53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.847 - 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.847 - 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $-0.847 - 0.531i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (643, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ -0.847 - 0.531i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2310846610 - 0.8036817022i\)
\(L(\frac12)\) \(\approx\) \(0.2310846610 - 0.8036817022i\)
\(L(1)\) \(\approx\) \(0.8719875469 - 0.2187257814i\)
\(L(1)\) \(\approx\) \(0.8719875469 - 0.2187257814i\)

Euler product

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

−18.05853242887688891833460124250, −17.52042276583092669505319977815, −16.63090181176627253145051671664, −15.87545221505111970196966158231, −15.37623335379288491080324362169, −14.63342607227431549037315564341, −14.20916315842818951846027862323, −13.38020181821310987608542517052, −12.68325769362609356929983069076, −12.18820806314367639142299933109, −11.037062423004831535671965375514, −10.81554319556212231366264750452, −10.06921013029502883138481946764, −9.5653743723899070087033385381, −8.41021019910559131551995036488, −7.94766424906655477533800622198, −7.21352942537455397036203705565, −6.58541583715858445190083383969, −5.71988041075183205740172702975, −5.24758279946250764198571012694, −4.229058891990290178911041906881, −3.44177891609666345538610858818, −2.654083333004592359561199077143, −2.260515275576287951020633679528, −1.009677959443021047760940582198, 0.23351473941402634467675849223, 1.17364024895978174794074245201, 2.16748820397718686509554255142, 2.706696372191601686895554112202, 3.85965795050855602465424198629, 4.52333764573638272857193100145, 5.243668687652735133814189554131, 5.63140098184324470325356629747, 6.74466998611935043438576312477, 7.448188122103983317058474205683, 8.01251650997610783082708989300, 8.82413042276430583557584985868, 9.63664560673051411221458065646, 9.74763647933665459071430762353, 10.92795987765058136461034723159, 11.64589455367992735467931418661, 12.08288941169042506360705943804, 12.911127418053875690093023667614, 13.5602535847762303013611900814, 13.883658021056179889453718594405, 14.9132721047456875953391151976, 15.66086320331646461869057685700, 16.12265636790529277281877742820, 16.682931720542915318934378049222, 17.432137288236481704285869462881

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