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Properties

Label 2-48-16.5-c5-0-4
Degree $2$
Conductor $48$
Sign $0.991 + 0.132i$
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  + (−4.89 − 2.83i)2-s + (−6.36 + 6.36i)3-s + (15.9 + 27.7i)4-s + (−71.9 − 71.9i)5-s + (49.1 − 13.1i)6-s + 158. i·7-s + (0.326 − 181. i)8-s − 81i·9-s + (148. + 556. i)10-s + (251. + 251. i)11-s + (−278. − 74.8i)12-s + (365. − 365. i)13-s + (449. − 777. i)14-s + 916.·15-s + (−514. + 885. i)16-s + 1.50e3·17-s + ⋯
L(s)  = 1  + (−0.865 − 0.500i)2-s + (−0.408 + 0.408i)3-s + (0.498 + 0.866i)4-s + (−1.28 − 1.28i)5-s + (0.557 − 0.149i)6-s + 1.22i·7-s + (0.00180 − 0.999i)8-s − 0.333i·9-s + (0.470 + 1.75i)10-s + (0.627 + 0.627i)11-s + (−0.557 − 0.150i)12-s + (0.600 − 0.600i)13-s + (0.613 − 1.06i)14-s + 1.05·15-s + (−0.502 + 0.864i)16-s + 1.26·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.132i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.991 + 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.991 + 0.132i$
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: \((2,\ 48,\ (\ :5/2),\ 0.991 + 0.132i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.759594 - 0.0507035i\)
\(L(\frac12)\) \(\approx\) \(0.759594 - 0.0507035i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (4.89 + 2.83i)T \)
3 \( 1 + (6.36 - 6.36i)T \)
good5 \( 1 + (71.9 + 71.9i)T + 3.12e3iT^{2} \)
7 \( 1 - 158. iT - 1.68e4T^{2} \)
11 \( 1 + (-251. - 251. i)T + 1.61e5iT^{2} \)
13 \( 1 + (-365. + 365. i)T - 3.71e5iT^{2} \)
17 \( 1 - 1.50e3T + 1.41e6T^{2} \)
19 \( 1 + (-1.79e3 + 1.79e3i)T - 2.47e6iT^{2} \)
23 \( 1 - 2.09e3iT - 6.43e6T^{2} \)
29 \( 1 + (2.34e3 - 2.34e3i)T - 2.05e7iT^{2} \)
31 \( 1 + 976.T + 2.86e7T^{2} \)
37 \( 1 + (-4.69e3 - 4.69e3i)T + 6.93e7iT^{2} \)
41 \( 1 + 1.43e4iT - 1.15e8T^{2} \)
43 \( 1 + (-3.69e3 - 3.69e3i)T + 1.47e8iT^{2} \)
47 \( 1 + 4.92e3T + 2.29e8T^{2} \)
53 \( 1 + (-5.19e3 - 5.19e3i)T + 4.18e8iT^{2} \)
59 \( 1 + (-1.64e4 - 1.64e4i)T + 7.14e8iT^{2} \)
61 \( 1 + (-1.75e4 + 1.75e4i)T - 8.44e8iT^{2} \)
67 \( 1 + (-2.87e4 + 2.87e4i)T - 1.35e9iT^{2} \)
71 \( 1 + 823. iT - 1.80e9T^{2} \)
73 \( 1 - 6.77e4iT - 2.07e9T^{2} \)
79 \( 1 - 4.77e4T + 3.07e9T^{2} \)
83 \( 1 + (-3.13e4 + 3.13e4i)T - 3.93e9iT^{2} \)
89 \( 1 - 2.43e4iT - 5.58e9T^{2} \)
97 \( 1 - 9.82e4T + 8.58e9T^{2} \)
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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

−15.32111143184462892697127619888, −12.77908078391333826855219491628, −11.99360307432456284275736073698, −11.38014705035528517646707932210, −9.544592143299401336034105744405, −8.751607931419586845753713399903, −7.55348522500824098223200642297, −5.27589603536956556479516978713, −3.57477042559442777119398140588, −0.955246705900861514917751876230, 0.833705461673835665508579021852, 3.68094592674337121663808572000, 6.24254046034457480446385689368, 7.28922165686740386503476460773, 8.018824779375850171931925336261, 10.07092779353512413870585583336, 11.08394147844308287020704985115, 11.78167591100800744144938438447, 14.04170489350671399858004336417, 14.65045078705014753088701477686

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