Properties

Label 2-448-28.19-c1-0-3
Degree $2$
Conductor $448$
Sign $0.444 - 0.895i$
Analytic cond. $3.57729$
Root an. cond. $1.89137$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 1.5i)3-s + (1.5 + 0.866i)5-s + (1.73 + 2i)7-s + (0.866 − 0.5i)11-s − 3.46i·13-s + 3i·15-s + (−1.5 + 0.866i)17-s + (−2.59 + 4.5i)19-s + (−1.50 + 4.33i)21-s + (−0.866 − 0.5i)23-s + (−1 − 1.73i)25-s + 5.19·27-s − 4·29-s + (0.866 + 1.5i)31-s + (1.5 + 0.866i)33-s + ⋯
L(s)  = 1  + (0.499 + 0.866i)3-s + (0.670 + 0.387i)5-s + (0.654 + 0.755i)7-s + (0.261 − 0.150i)11-s − 0.960i·13-s + 0.774i·15-s + (−0.363 + 0.210i)17-s + (−0.596 + 1.03i)19-s + (−0.327 + 0.944i)21-s + (−0.180 − 0.104i)23-s + (−0.200 − 0.346i)25-s + 1.00·27-s − 0.742·29-s + (0.155 + 0.269i)31-s + (0.261 + 0.150i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.444 - 0.895i$
Analytic conductor: \(3.57729\)
Root analytic conductor: \(1.89137\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1/2),\ 0.444 - 0.895i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.63779 + 1.01609i\)
\(L(\frac12)\) \(\approx\) \(1.63779 + 1.01609i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.73 - 2i)T \)
good3 \( 1 + (-0.866 - 1.5i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.5 - 0.866i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.866 + 0.5i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.46iT - 13T^{2} \)
17 \( 1 + (1.5 - 0.866i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.59 - 4.5i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.866 + 0.5i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + (-0.866 - 1.5i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.5 + 2.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 3.46iT - 41T^{2} \)
43 \( 1 + 2iT - 43T^{2} \)
47 \( 1 + (4.33 - 7.5i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.59 + 4.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.5 - 2.59i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.59 + 1.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 14iT - 71T^{2} \)
73 \( 1 + (-7.5 + 4.33i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.79 + 4.5i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 13.8T + 83T^{2} \)
89 \( 1 + (-13.5 - 7.79i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 17.3iT - 97T^{2} \)
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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

−10.97517590636691842209829807427, −10.29233125531613752785678339070, −9.502844452221826321801328251494, −8.670676711468156846644875228916, −7.87254139132288101596382382545, −6.36841696237561543730530495532, −5.57355300128677886401587579543, −4.38082759710359553075574900837, −3.24354565953934307254864852773, −2.00381032521859041150670228900, 1.39876642209692133624301680685, 2.31965705615484027027562375473, 4.14129297242995947238260935568, 5.10412713105915544256972825094, 6.56208256735920456303519445966, 7.20488959036581092647778589963, 8.178879946103446912780320300217, 9.033770817009944580319910657090, 9.920592196395164269045865155912, 11.06751089746579069021160570858

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