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Properties

Label 2-48-16.13-c5-0-9
Degree $2$
Conductor $48$
Sign $0.840 - 0.541i$
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  + (−5.59 − 0.853i)2-s + (6.36 + 6.36i)3-s + (30.5 + 9.54i)4-s + (16.4 − 16.4i)5-s + (−30.1 − 41.0i)6-s + 18.3i·7-s + (−162. − 79.4i)8-s + 81i·9-s + (−105. + 77.9i)10-s + (92.4 − 92.4i)11-s + (133. + 255. i)12-s + (605. + 605. i)13-s + (15.6 − 102. i)14-s + 209.·15-s + (841. + 583. i)16-s + 1.23e3·17-s + ⋯
L(s)  = 1  + (−0.988 − 0.150i)2-s + (0.408 + 0.408i)3-s + (0.954 + 0.298i)4-s + (0.294 − 0.294i)5-s + (−0.341 − 0.465i)6-s + 0.141i·7-s + (−0.898 − 0.438i)8-s + 0.333i·9-s + (−0.335 + 0.246i)10-s + (0.230 − 0.230i)11-s + (0.267 + 0.511i)12-s + (0.993 + 0.993i)13-s + (0.0213 − 0.140i)14-s + 0.240·15-s + (0.822 + 0.569i)16-s + 1.03·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.840 - 0.541i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.840 - 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.840 - 0.541i$
Analytic conductor: \(7.69842\)
Root analytic conductor: \(2.77460\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :5/2),\ 0.840 - 0.541i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.28740 + 0.378815i\)
\(L(\frac12)\) \(\approx\) \(1.28740 + 0.378815i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.59 + 0.853i)T \)
3 \( 1 + (-6.36 - 6.36i)T \)
good5 \( 1 + (-16.4 + 16.4i)T - 3.12e3iT^{2} \)
7 \( 1 - 18.3iT - 1.68e4T^{2} \)
11 \( 1 + (-92.4 + 92.4i)T - 1.61e5iT^{2} \)
13 \( 1 + (-605. - 605. i)T + 3.71e5iT^{2} \)
17 \( 1 - 1.23e3T + 1.41e6T^{2} \)
19 \( 1 + (-1.37e3 - 1.37e3i)T + 2.47e6iT^{2} \)
23 \( 1 - 130. iT - 6.43e6T^{2} \)
29 \( 1 + (675. + 675. i)T + 2.05e7iT^{2} \)
31 \( 1 + 2.82e3T + 2.86e7T^{2} \)
37 \( 1 + (-7.15e3 + 7.15e3i)T - 6.93e7iT^{2} \)
41 \( 1 + 4.04e3iT - 1.15e8T^{2} \)
43 \( 1 + (5.21e3 - 5.21e3i)T - 1.47e8iT^{2} \)
47 \( 1 + 6.44e3T + 2.29e8T^{2} \)
53 \( 1 + (-1.93e4 + 1.93e4i)T - 4.18e8iT^{2} \)
59 \( 1 + (3.50e4 - 3.50e4i)T - 7.14e8iT^{2} \)
61 \( 1 + (3.41e4 + 3.41e4i)T + 8.44e8iT^{2} \)
67 \( 1 + (2.78e4 + 2.78e4i)T + 1.35e9iT^{2} \)
71 \( 1 + 8.17e3iT - 1.80e9T^{2} \)
73 \( 1 + 3.88e4iT - 2.07e9T^{2} \)
79 \( 1 - 2.05e4T + 3.07e9T^{2} \)
83 \( 1 + (3.97e4 + 3.97e4i)T + 3.93e9iT^{2} \)
89 \( 1 - 6.27e4iT - 5.58e9T^{2} \)
97 \( 1 - 1.57e4T + 8.58e9T^{2} \)
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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

−14.94017119211343749940247588123, −13.70289980307422356683704535211, −12.14287982784170102190372741538, −11.00133190116521119517760275121, −9.682659640943019943319150742479, −8.896214734358742092332937711129, −7.60989495993233633526929103742, −5.88385073982909766967597582186, −3.52263342308573639088503460453, −1.51263638588256313839933226244, 1.09153712226307085179824656392, 2.97824967827142622284573165336, 5.89028329966824401131898171620, 7.24060529423094983475298262542, 8.343504756851586620489480062335, 9.624401659455001572446195474500, 10.72749211378391761066982412208, 12.05204092346259300041263766742, 13.50003203544009654466572370476, 14.71743667547037522721353826638

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