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Properties

Label 2-448-56.11-c2-0-12
Degree $2$
Conductor $448$
Sign $-0.728 - 0.684i$
Analytic cond. $12.2071$
Root an. cond. $3.49386$
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  + (−2.20 + 3.81i)3-s + (3.13 − 1.80i)5-s + (4.05 + 5.70i)7-s + (−5.18 − 8.98i)9-s + (−1.85 + 3.21i)11-s + 15.5i·13-s + 15.9i·15-s + (14.3 − 24.7i)17-s + (15.0 + 25.9i)19-s + (−30.6 + 2.89i)21-s + (11.3 − 6.52i)23-s + (−5.95 + 10.3i)25-s + 6.04·27-s + 8.55i·29-s + (−13.2 − 7.63i)31-s + ⋯
L(s)  = 1  + (−0.733 + 1.27i)3-s + (0.626 − 0.361i)5-s + (0.579 + 0.815i)7-s + (−0.576 − 0.998i)9-s + (−0.168 + 0.291i)11-s + 1.19i·13-s + 1.06i·15-s + (0.841 − 1.45i)17-s + (0.789 + 1.36i)19-s + (−1.46 + 0.138i)21-s + (0.491 − 0.283i)23-s + (−0.238 + 0.412i)25-s + 0.224·27-s + 0.294i·29-s + (−0.426 − 0.246i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.728 - 0.684i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.728 - 0.684i$
Analytic conductor: \(12.2071\)
Root analytic conductor: \(3.49386\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1),\ -0.728 - 0.684i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.398961144\)
\(L(\frac12)\) \(\approx\) \(1.398961144\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-4.05 - 5.70i)T \)
good3 \( 1 + (2.20 - 3.81i)T + (-4.5 - 7.79i)T^{2} \)
5 \( 1 + (-3.13 + 1.80i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (1.85 - 3.21i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 15.5iT - 169T^{2} \)
17 \( 1 + (-14.3 + 24.7i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (-15.0 - 25.9i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-11.3 + 6.52i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 - 8.55iT - 841T^{2} \)
31 \( 1 + (13.2 + 7.63i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (30.2 - 17.4i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 0.875T + 1.68e3T^{2} \)
43 \( 1 + 33.4T + 1.84e3T^{2} \)
47 \( 1 + (76.1 - 43.9i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (19.2 + 11.0i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-48.9 + 84.8i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (34.7 - 20.0i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (7.33 - 12.6i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 112. iT - 5.04e3T^{2} \)
73 \( 1 + (-56.4 + 97.7i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-56.0 + 32.3i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 58.2T + 6.88e3T^{2} \)
89 \( 1 + (22.2 + 38.5i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 166.T + 9.40e3T^{2} \)
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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

−11.44348697270740448307191826858, −10.09432356925325500661363258464, −9.613497965279943100607610213178, −8.903653053983243190149848905906, −7.59977781396327543202033595774, −6.19845408488525725926908693112, −5.15403280523464268004435257796, −4.92058908513760900929133551390, −3.42837117972417572522311060992, −1.71618797017997075737909795060, 0.66609803540593898108889646985, 1.78076235457640139951874325471, 3.33900621371505590126566044087, 5.12431702080666969136151685640, 5.88967667666993389900413644296, 6.84321858636534321859811027133, 7.60456104174071360356033508088, 8.391609379077000546217121850678, 9.951696358328636935834007725647, 10.68958509903989261916507780282

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