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Properties

Label 2-48-48.5-c2-0-1
Degree $2$
Conductor $48$
Sign $-0.0690 - 0.997i$
Analytic cond. $1.30790$
Root an. cond. $1.14363$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.28 − 1.53i)2-s + (−2.06 + 2.17i)3-s + (−0.697 + 3.93i)4-s + (−3.17 + 3.17i)5-s + (5.98 + 0.363i)6-s + 6.03i·7-s + (6.93 − 3.99i)8-s + (−0.485 − 8.98i)9-s + (8.95 + 0.787i)10-s + (−13.0 + 13.0i)11-s + (−7.13 − 9.64i)12-s + (6.39 − 6.39i)13-s + (9.25 − 7.76i)14-s + (−0.363 − 13.4i)15-s + (−15.0 − 5.49i)16-s − 4.39i·17-s + ⋯
L(s)  = 1  + (−0.642 − 0.766i)2-s + (−0.687 + 0.725i)3-s + (−0.174 + 0.984i)4-s + (−0.635 + 0.635i)5-s + (0.998 + 0.0606i)6-s + 0.862i·7-s + (0.866 − 0.499i)8-s + (−0.0539 − 0.998i)9-s + (0.895 + 0.0787i)10-s + (−1.18 + 1.18i)11-s + (−0.594 − 0.803i)12-s + (0.491 − 0.491i)13-s + (0.661 − 0.554i)14-s + (−0.0242 − 0.898i)15-s + (−0.939 − 0.343i)16-s − 0.258i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0690 - 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0690 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.0690 - 0.997i$
Analytic conductor: \(1.30790\)
Root analytic conductor: \(1.14363\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :1),\ -0.0690 - 0.997i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.316712 + 0.339379i\)
\(L(\frac12)\) \(\approx\) \(0.316712 + 0.339379i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.28 + 1.53i)T \)
3 \( 1 + (2.06 - 2.17i)T \)
good5 \( 1 + (3.17 - 3.17i)T - 25iT^{2} \)
7 \( 1 - 6.03iT - 49T^{2} \)
11 \( 1 + (13.0 - 13.0i)T - 121iT^{2} \)
13 \( 1 + (-6.39 + 6.39i)T - 169iT^{2} \)
17 \( 1 + 4.39iT - 289T^{2} \)
19 \( 1 + (3.21 - 3.21i)T - 361iT^{2} \)
23 \( 1 - 34.0T + 529T^{2} \)
29 \( 1 + (-27.9 - 27.9i)T + 841iT^{2} \)
31 \( 1 + 7.90T + 961T^{2} \)
37 \( 1 + (-20.0 - 20.0i)T + 1.36e3iT^{2} \)
41 \( 1 + 45.1T + 1.68e3T^{2} \)
43 \( 1 + (36.0 + 36.0i)T + 1.84e3iT^{2} \)
47 \( 1 - 5.08iT - 2.20e3T^{2} \)
53 \( 1 + (-20.7 + 20.7i)T - 2.80e3iT^{2} \)
59 \( 1 + (39.0 - 39.0i)T - 3.48e3iT^{2} \)
61 \( 1 + (49.8 - 49.8i)T - 3.72e3iT^{2} \)
67 \( 1 + (-44.9 + 44.9i)T - 4.48e3iT^{2} \)
71 \( 1 - 46.6T + 5.04e3T^{2} \)
73 \( 1 - 97.3iT - 5.32e3T^{2} \)
79 \( 1 + 40.1T + 6.24e3T^{2} \)
83 \( 1 + (35.5 + 35.5i)T + 6.88e3iT^{2} \)
89 \( 1 - 69.6T + 7.92e3T^{2} \)
97 \( 1 - 61.0T + 9.40e3T^{2} \)
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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

−15.65061818658617117837545529611, −15.13317440268079478791160059939, −12.90354193399363483981995520109, −11.94010840671461240402892333882, −10.90286476644049839051916912929, −10.08381540031382903958577855056, −8.700524745389630130289362647638, −7.13956262265056961458114452840, −4.97537017581361145804958518690, −3.10261436760194827133415291207, 0.65516185204374758761459669322, 4.87423067288907793656493207716, 6.35421624668009846040705573222, 7.68419483407621562212922767437, 8.548584754704677976921612408905, 10.51174188168293971656189532604, 11.36195249526873254692522054352, 13.07139933918106138215869217070, 13.84679467885349073723410275426, 15.61518075999671282365411086830

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