Properties

Label 2-448-28.3-c1-0-1
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: $\chi_{448} (255, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1/2),\ -0.444 - 0.895i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.552390 + 0.890374i\)
\(L(\frac12)\) \(\approx\) \(0.552390 + 0.890374i\)
\(L(\frac{3}{2})\) 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

−11.31720247659013606837987464450, −10.39015510085381398581691672600, −9.473081664089922579122779762346, −9.177487056622605123152066315929, −7.79221747629882564762322394287, −6.42607145732180434862019494542, −5.59826262745668152368620139226, −4.82797496862686975192908419961, −3.56435278860499199889560877490, −1.99357925549262161743765988458, 0.69114383661100117430747661149, 2.38013615870705240791986313311, 3.74896135017236353891619496470, 5.34367266927238720263413611212, 6.24835943678682912895868370976, 7.02129210612622412772528501440, 7.70278037159101075794762228072, 9.159865080731456151786160698726, 10.03686834106007009424683312008, 10.76301899368026729488628030259

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