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Properties

Label 1-48-48.35-r0-0-0
Degree $1$
Conductor $48$
Sign $0.923 + 0.382i$
Analytic cond. $0.222911$
Root an. cond. $0.222911$
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  + i·5-s + 7-s − i·11-s + i·13-s − 17-s − i·19-s − 23-s − 25-s − i·29-s − 31-s + i·35-s − i·37-s + 41-s + i·43-s + 47-s + ⋯
L(s)  = 1  + i·5-s + 7-s − i·11-s + i·13-s − 17-s − i·19-s − 23-s − 25-s − i·29-s − 31-s + i·35-s − i·37-s + 41-s + i·43-s + 47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.923 + 0.382i$
Analytic conductor: \(0.222911\)
Root analytic conductor: \(0.222911\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 48,\ (0:\ ),\ 0.923 + 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8717045904 + 0.1733928237i\)
\(L(\frac12)\) \(\approx\) \(0.8717045904 + 0.1733928237i\)
\(L(1)\) \(\approx\) \(1.020168853 + 0.1259994663i\)
\(L(1)\) \(\approx\) \(1.020168853 + 0.1259994663i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
13 \( 1 + iT \)
17 \( 1 \)
19 \( 1 + T \)
23 \( 1 \)
29 \( 1 \)
31 \( 1 \)
37 \( 1 - iT \)
41 \( 1 \)
43 \( 1 + iT \)
47 \( 1 \)
53 \( 1 \)
59 \( 1 \)
61 \( 1 - T \)
67 \( 1 \)
71 \( 1 - iT \)
73 \( 1 \)
79 \( 1 \)
83 \( 1 \)
89 \( 1 - T \)
97 \( 1 \)
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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

−33.70967941767759775557811727044, −32.82592496398376289645925692085, −31.52053978295450363195863858800, −30.59570491619174589522255379120, −29.197281748281395876095205863079, −27.95578674194250931122874826180, −27.28504208150445842359856195294, −25.54378180805934922900677470957, −24.54836107410118062402862288257, −23.55144697311556623809667657457, −22.09933021448122534253160481272, −20.599348196665348037922352600946, −20.10826766858780377225970176470, −18.12603892526519624278503613558, −17.26348221080926405546689864202, −15.81829999763547334524480872993, −14.5610912721199171472968092355, −13.00822370664870217579803931959, −11.93494806825236707414784471852, −10.33389662102123585133413954652, −8.7659999755719292502196316342, −7.610903846836790593628450221260, −5.50451924769678764880895326870, −4.28674929978863646743381271242, −1.781012879698985036991061601368, 2.3034635391148037906869057256, 4.19414022755256984159923204976, 6.083971482044770317661908398033, 7.50958020377302481533582415431, 9.02495658118244187700793674788, 10.85584748909344450820592331202, 11.55299350201086034274972350221, 13.643666525876531529129030505082, 14.53786493282839845601741011695, 15.87836807923724293387222796553, 17.49379824529992687770081326112, 18.49602924764055737428529132403, 19.6880634972816490388960889348, 21.33893062000594098001209159786, 22.08599771811400200179247418824, 23.64976436104949759960903966951, 24.53866485801569517424613073297, 26.23834880429563446933101756048, 26.83860122179109652950169004029, 28.24025099518263954230937623053, 29.61489387186538137389796403422, 30.56630792509172516373790244024, 31.52782831194015653545716395537, 33.1064569788815915340320970165, 34.07157613187348546686696076571

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