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Properties

Label 2-48-16.3-c6-0-15
Degree $2$
Conductor $48$
Sign $-0.999 - 0.00916i$
Analytic cond. $11.0425$
Root an. cond. $3.32304$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.75 − 7.06i)2-s + (−11.0 + 11.0i)3-s + (−35.8 + 53.0i)4-s + (107. − 107. i)5-s + (119. + 36.4i)6-s + 5.36·7-s + (509. + 53.6i)8-s − 242. i·9-s + (−1.16e3 − 356. i)10-s + (−48.9 − 48.9i)11-s + (−190. − 979. i)12-s + (−2.11e3 − 2.11e3i)13-s + (−20.1 − 37.9i)14-s + 2.37e3i·15-s + (−1.53e3 − 3.79e3i)16-s − 3.88e3·17-s + ⋯
L(s)  = 1  + (−0.469 − 0.882i)2-s + (−0.408 + 0.408i)3-s + (−0.559 + 0.828i)4-s + (0.862 − 0.862i)5-s + (0.552 + 0.168i)6-s + 0.0156·7-s + (0.994 + 0.104i)8-s − 0.333i·9-s + (−1.16 − 0.356i)10-s + (−0.0367 − 0.0367i)11-s + (−0.110 − 0.566i)12-s + (−0.964 − 0.964i)13-s + (−0.00734 − 0.0138i)14-s + 0.704i·15-s + (−0.374 − 0.927i)16-s − 0.790·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00916i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.999 - 0.00916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(48\)    =    \(2^{4} \cdot 3\)
Sign: $-0.999 - 0.00916i$
Analytic conductor: \(11.0425\)
Root analytic conductor: \(3.32304\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{48} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 48,\ (\ :3),\ -0.999 - 0.00916i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.00265107 + 0.578264i\)
\(L(\frac12)\) \(\approx\) \(0.00265107 + 0.578264i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.75 + 7.06i)T \)
3 \( 1 + (11.0 - 11.0i)T \)
good5 \( 1 + (-107. + 107. i)T - 1.56e4iT^{2} \)
7 \( 1 - 5.36T + 1.17e5T^{2} \)
11 \( 1 + (48.9 + 48.9i)T + 1.77e6iT^{2} \)
13 \( 1 + (2.11e3 + 2.11e3i)T + 4.82e6iT^{2} \)
17 \( 1 + 3.88e3T + 2.41e7T^{2} \)
19 \( 1 + (5.87e3 - 5.87e3i)T - 4.70e7iT^{2} \)
23 \( 1 + 3.23e3T + 1.48e8T^{2} \)
29 \( 1 + (3.05e4 + 3.05e4i)T + 5.94e8iT^{2} \)
31 \( 1 + 5.40e4iT - 8.87e8T^{2} \)
37 \( 1 + (-3.36e4 + 3.36e4i)T - 2.56e9iT^{2} \)
41 \( 1 - 2.90e4iT - 4.75e9T^{2} \)
43 \( 1 + (2.45e4 + 2.45e4i)T + 6.32e9iT^{2} \)
47 \( 1 - 5.81e4iT - 1.07e10T^{2} \)
53 \( 1 + (-2.70e4 + 2.70e4i)T - 2.21e10iT^{2} \)
59 \( 1 + (-2.25e5 - 2.25e5i)T + 4.21e10iT^{2} \)
61 \( 1 + (1.78e3 + 1.78e3i)T + 5.15e10iT^{2} \)
67 \( 1 + (-1.22e5 + 1.22e5i)T - 9.04e10iT^{2} \)
71 \( 1 + 2.13e5T + 1.28e11T^{2} \)
73 \( 1 + 3.37e5iT - 1.51e11T^{2} \)
79 \( 1 - 7.15e5iT - 2.43e11T^{2} \)
83 \( 1 + (-3.38e5 + 3.38e5i)T - 3.26e11iT^{2} \)
89 \( 1 - 5.27e5iT - 4.96e11T^{2} \)
97 \( 1 + 1.55e6T + 8.32e11T^{2} \)
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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

−13.31407181952624280055151414230, −12.64385265438087137114365381351, −11.34909829749610797705582316887, −10.07975200343480304885162213802, −9.357138925543055274198150233138, −7.961313845721206889789798024209, −5.69045198860265411885906284949, −4.28121400940629378467690133315, −2.14316565175096654579010401326, −0.30451656797499915339183431681, 1.99266225469216089923771525447, 4.95212711722654074897966724556, 6.47764585598543101003517648121, 7.08583686516146085665242108493, 8.872868852454238129739518790634, 10.09445690546467540234046044810, 11.16801529787343944286888896485, 12.97588534983099232288242940356, 14.10896095596169648892957481041, 14.86892201647309691328657185132

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