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Properties

Label 2-48-48.11-c1-0-4
Degree $2$
Conductor $48$
Sign $0.955 + 0.296i$
Analytic cond. $0.383281$
Root an. cond. $0.619097$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.39 − 0.204i)2-s + (−1.52 − 0.814i)3-s + (1.91 − 0.573i)4-s + (−2.08 + 2.08i)5-s + (−2.30 − 0.826i)6-s − 1.14·7-s + (2.56 − 1.19i)8-s + (1.67 + 2.48i)9-s + (−2.48 + 3.34i)10-s + (−1.67 − 1.67i)11-s + (−3.39 − 0.683i)12-s + (0.146 − 0.146i)13-s + (−1.60 + 0.234i)14-s + (4.88 − 1.48i)15-s + (3.34 − 2.19i)16-s − 5.59i·17-s + ⋯
L(s)  = 1  + (0.989 − 0.144i)2-s + (−0.882 − 0.470i)3-s + (0.958 − 0.286i)4-s + (−0.931 + 0.931i)5-s + (−0.941 − 0.337i)6-s − 0.433·7-s + (0.906 − 0.422i)8-s + (0.558 + 0.829i)9-s + (−0.787 + 1.05i)10-s + (−0.504 − 0.504i)11-s + (−0.980 − 0.197i)12-s + (0.0405 − 0.0405i)13-s + (−0.428 + 0.0627i)14-s + (1.26 − 0.384i)15-s + (0.835 − 0.549i)16-s − 1.35i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.296i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 + 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.955 + 0.296i$
Analytic conductor: \(0.383281\)
Root analytic conductor: \(0.619097\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :1/2),\ 0.955 + 0.296i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.969872 - 0.146932i\)
\(L(\frac12)\) \(\approx\) \(0.969872 - 0.146932i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 + 0.204i)T \)
3 \( 1 + (1.52 + 0.814i)T \)
good5 \( 1 + (2.08 - 2.08i)T - 5iT^{2} \)
7 \( 1 + 1.14T + 7T^{2} \)
11 \( 1 + (1.67 + 1.67i)T + 11iT^{2} \)
13 \( 1 + (-0.146 + 0.146i)T - 13iT^{2} \)
17 \( 1 + 5.59iT - 17T^{2} \)
19 \( 1 + (-1.48 - 1.48i)T + 19iT^{2} \)
23 \( 1 - 3.34iT - 23T^{2} \)
29 \( 1 + (-3.51 - 3.51i)T + 29iT^{2} \)
31 \( 1 - 5.83iT - 31T^{2} \)
37 \( 1 + (4.83 + 4.83i)T + 37iT^{2} \)
41 \( 1 + 0.610T + 41T^{2} \)
43 \( 1 + (1.48 - 1.48i)T - 43iT^{2} \)
47 \( 1 + 6.41T + 47T^{2} \)
53 \( 1 + (0.164 - 0.164i)T - 53iT^{2} \)
59 \( 1 + (-9.05 - 9.05i)T + 59iT^{2} \)
61 \( 1 + (-4.53 + 4.53i)T - 61iT^{2} \)
67 \( 1 + (0.635 + 0.635i)T + 67iT^{2} \)
71 \( 1 + 6.90iT - 71T^{2} \)
73 \( 1 - 7.07iT - 73T^{2} \)
79 \( 1 + 9.83iT - 79T^{2} \)
83 \( 1 + (-8.09 + 8.09i)T - 83iT^{2} \)
89 \( 1 + 0.490T + 89T^{2} \)
97 \( 1 - 12.3T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.84100511534620690586494358379, −14.36259268249845167343489278679, −13.28015389622210288670725243095, −12.05532702744457275670804478033, −11.31559656175264282170099660754, −10.34445080935325771866034687834, −7.55822831807466088875101804004, −6.67885022596155699292730954190, −5.18907827279778118711461963409, −3.23986236753145177440224526434, 3.96912143353736020292877361809, 5.05116096829299101764919662120, 6.53243456904091180036656137937, 8.159555197391032021727501905818, 10.14177829328048534392621528298, 11.45415904042625078226026699297, 12.37809492195484728965615268727, 13.08980780372922882952397237629, 14.98475045464726205709953997993, 15.74531955652831910894806347706

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