Properties

Label 2-48-16.5-c5-0-11
Degree $2$
Conductor $48$
Sign $-0.457 + 0.889i$
Analytic cond. $7.69842$
Root an. cond. $2.77460$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.04 − 4.76i)2-s + (−6.36 + 6.36i)3-s + (−13.4 + 29.0i)4-s + (25.2 + 25.2i)5-s + (49.7 + 10.9i)6-s − 124. i·7-s + (179. − 24.3i)8-s − 81i·9-s + (43.5 − 197. i)10-s + (−241. − 241. i)11-s + (−99.1 − 270. i)12-s + (215. − 215. i)13-s + (−593. + 379. i)14-s − 321.·15-s + (−662. − 781. i)16-s − 1.34e3·17-s + ⋯
L(s)  = 1  + (−0.538 − 0.842i)2-s + (−0.408 + 0.408i)3-s + (−0.420 + 0.907i)4-s + (0.452 + 0.452i)5-s + (0.563 + 0.124i)6-s − 0.960i·7-s + (0.990 − 0.134i)8-s − 0.333i·9-s + (0.137 − 0.624i)10-s + (−0.600 − 0.600i)11-s + (−0.198 − 0.542i)12-s + (0.353 − 0.353i)13-s + (−0.809 + 0.517i)14-s − 0.369·15-s + (−0.646 − 0.762i)16-s − 1.12·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.457 + 0.889i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.457 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.457 + 0.889i$
Analytic conductor: \(7.69842\)
Root analytic conductor: \(2.77460\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :5/2),\ -0.457 + 0.889i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.443338 - 0.726435i\)
\(L(\frac12)\) \(\approx\) \(0.443338 - 0.726435i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.04 + 4.76i)T \)
3 \( 1 + (6.36 - 6.36i)T \)
good5 \( 1 + (-25.2 - 25.2i)T + 3.12e3iT^{2} \)
7 \( 1 + 124. iT - 1.68e4T^{2} \)
11 \( 1 + (241. + 241. i)T + 1.61e5iT^{2} \)
13 \( 1 + (-215. + 215. i)T - 3.71e5iT^{2} \)
17 \( 1 + 1.34e3T + 1.41e6T^{2} \)
19 \( 1 + (-1.85e3 + 1.85e3i)T - 2.47e6iT^{2} \)
23 \( 1 + 2.91e3iT - 6.43e6T^{2} \)
29 \( 1 + (-1.12e3 + 1.12e3i)T - 2.05e7iT^{2} \)
31 \( 1 - 4.21e3T + 2.86e7T^{2} \)
37 \( 1 + (-3.90e3 - 3.90e3i)T + 6.93e7iT^{2} \)
41 \( 1 - 1.42e4iT - 1.15e8T^{2} \)
43 \( 1 + (1.12e4 + 1.12e4i)T + 1.47e8iT^{2} \)
47 \( 1 + 1.30e4T + 2.29e8T^{2} \)
53 \( 1 + (9.96e3 + 9.96e3i)T + 4.18e8iT^{2} \)
59 \( 1 + (5.31e3 + 5.31e3i)T + 7.14e8iT^{2} \)
61 \( 1 + (1.72e4 - 1.72e4i)T - 8.44e8iT^{2} \)
67 \( 1 + (-3.78e4 + 3.78e4i)T - 1.35e9iT^{2} \)
71 \( 1 - 2.60e4iT - 1.80e9T^{2} \)
73 \( 1 - 2.10e4iT - 2.07e9T^{2} \)
79 \( 1 + 9.53e4T + 3.07e9T^{2} \)
83 \( 1 + (-6.60e3 + 6.60e3i)T - 3.93e9iT^{2} \)
89 \( 1 - 1.32e5iT - 5.58e9T^{2} \)
97 \( 1 - 7.39e4T + 8.58e9T^{2} \)
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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

−13.88273585697913691045982125058, −13.14379461645415841973670693588, −11.47265100351022453658017171318, −10.67653157736308708096071211688, −9.836959512879220870961905130252, −8.336713591286851432431432112971, −6.71436182051258440847610971562, −4.61813729833095365538998511506, −2.90600699167472917888002505509, −0.57568514755893846409116640694, 1.66677508781926504777641526113, 5.08143174957016234890660757690, 6.06511308557390810045166195032, 7.53308666987786419478320169348, 8.857983286420991266942241008056, 9.911193210394541381578750544489, 11.49287745492879342428600459957, 12.88691110190613665129550889311, 13.92721383528881083948282484286, 15.36317344228687236262957127772

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