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Properties

Label 2-48-16.5-c5-0-13
Degree $2$
Conductor $48$
Sign $0.425 + 0.904i$
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  + (0.0662 + 5.65i)2-s + (6.36 − 6.36i)3-s + (−31.9 + 0.749i)4-s + (−23.0 − 23.0i)5-s + (36.4 + 35.5i)6-s − 103. i·7-s + (−6.35 − 180. i)8-s − 81i·9-s + (129. − 132. i)10-s + (−300. − 300. i)11-s + (−198. + 208. i)12-s + (142. − 142. i)13-s + (587. − 6.87i)14-s − 293.·15-s + (1.02e3 − 47.9i)16-s − 12.5·17-s + ⋯
L(s)  = 1  + (0.0117 + 0.999i)2-s + (0.408 − 0.408i)3-s + (−0.999 + 0.0234i)4-s + (−0.413 − 0.413i)5-s + (0.413 + 0.403i)6-s − 0.801i·7-s + (−0.0351 − 0.999i)8-s − 0.333i·9-s + (0.408 − 0.417i)10-s + (−0.748 − 0.748i)11-s + (−0.398 + 0.417i)12-s + (0.234 − 0.234i)13-s + (0.800 − 0.00937i)14-s − 0.337·15-s + (0.998 − 0.0468i)16-s − 0.0105·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.425 + 0.904i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.425 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $0.425 + 0.904i$
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: \((2,\ 48,\ (\ :5/2),\ 0.425 + 0.904i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.945836 - 0.600439i\)
\(L(\frac12)\) \(\approx\) \(0.945836 - 0.600439i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0662 - 5.65i)T \)
3 \( 1 + (-6.36 + 6.36i)T \)
good5 \( 1 + (23.0 + 23.0i)T + 3.12e3iT^{2} \)
7 \( 1 + 103. iT - 1.68e4T^{2} \)
11 \( 1 + (300. + 300. i)T + 1.61e5iT^{2} \)
13 \( 1 + (-142. + 142. i)T - 3.71e5iT^{2} \)
17 \( 1 + 12.5T + 1.41e6T^{2} \)
19 \( 1 + (-844. + 844. i)T - 2.47e6iT^{2} \)
23 \( 1 - 335. iT - 6.43e6T^{2} \)
29 \( 1 + (5.86e3 - 5.86e3i)T - 2.05e7iT^{2} \)
31 \( 1 + 2.54e3T + 2.86e7T^{2} \)
37 \( 1 + (1.01e4 + 1.01e4i)T + 6.93e7iT^{2} \)
41 \( 1 - 384. iT - 1.15e8T^{2} \)
43 \( 1 + (-7.52e3 - 7.52e3i)T + 1.47e8iT^{2} \)
47 \( 1 - 1.53e4T + 2.29e8T^{2} \)
53 \( 1 + (1.62e4 + 1.62e4i)T + 4.18e8iT^{2} \)
59 \( 1 + (-2.73e4 - 2.73e4i)T + 7.14e8iT^{2} \)
61 \( 1 + (-1.12e4 + 1.12e4i)T - 8.44e8iT^{2} \)
67 \( 1 + (-4.97e4 + 4.97e4i)T - 1.35e9iT^{2} \)
71 \( 1 - 3.48e4iT - 1.80e9T^{2} \)
73 \( 1 - 9.34e3iT - 2.07e9T^{2} \)
79 \( 1 - 6.86e4T + 3.07e9T^{2} \)
83 \( 1 + (-4.37e4 + 4.37e4i)T - 3.93e9iT^{2} \)
89 \( 1 - 6.69e4iT - 5.58e9T^{2} \)
97 \( 1 - 1.63e5T + 8.58e9T^{2} \)
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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

−14.39169064226800418057629911427, −13.50143074612548007030425975266, −12.60862528132164169307412281392, −10.76439361278459212330219798790, −9.126318106418034887866472472095, −8.024773884219686786277348534790, −7.08158579614715621789236798216, −5.39218488185707811323413191917, −3.69480403596615584532275731946, −0.56153221724824368393612357742, 2.24457673109831234119900689247, 3.71392880402030964225204061774, 5.32787075265875864270240375861, 7.74279974872631727669915962956, 9.070523154796700359506875919933, 10.13790552563568806849522865366, 11.31957248250537812718042671022, 12.38217516285718005056268026438, 13.58730131583615841101145379404, 14.83786035253757642405754176937

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