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Properties

Label 2-448-28.19-c3-0-28
Degree $2$
Conductor $448$
Sign $0.829 + 0.559i$
Analytic cond. $26.4328$
Root an. cond. $5.14128$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.67 − 2.89i)3-s + (15.9 + 9.21i)5-s + (15.4 − 10.1i)7-s + (7.91 − 13.7i)9-s + (35.6 − 20.5i)11-s + 24.8i·13-s − 61.5i·15-s + (−41.3 + 23.8i)17-s + (30.9 − 53.5i)19-s + (−55.2 − 27.8i)21-s + (64.3 + 37.1i)23-s + (107. + 185. i)25-s − 143.·27-s − 28.8·29-s + (−11.5 − 20.0i)31-s + ⋯
L(s)  = 1  + (−0.321 − 0.557i)3-s + (1.42 + 0.824i)5-s + (0.836 − 0.548i)7-s + (0.293 − 0.507i)9-s + (0.976 − 0.563i)11-s + 0.530i·13-s − 1.06i·15-s + (−0.590 + 0.340i)17-s + (0.373 − 0.646i)19-s + (−0.574 − 0.289i)21-s + (0.583 + 0.337i)23-s + (0.858 + 1.48i)25-s − 1.02·27-s − 0.184·29-s + (−0.0670 − 0.116i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.829 + 0.559i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.829 + 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.829 + 0.559i$
Analytic conductor: \(26.4328\)
Root analytic conductor: \(5.14128\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :3/2),\ 0.829 + 0.559i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.732970502\)
\(L(\frac12)\) \(\approx\) \(2.732970502\)
\(L(\frac{5}{2})\) 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 \( 1 + (-15.4 + 10.1i)T \)
good3 \( 1 + (1.67 + 2.89i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (-15.9 - 9.21i)T + (62.5 + 108. i)T^{2} \)
11 \( 1 + (-35.6 + 20.5i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 24.8iT - 2.19e3T^{2} \)
17 \( 1 + (41.3 - 23.8i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-30.9 + 53.5i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-64.3 - 37.1i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 28.8T + 2.43e4T^{2} \)
31 \( 1 + (11.5 + 20.0i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (51.8 - 89.8i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 96.1iT - 6.89e4T^{2} \)
43 \( 1 - 195. iT - 7.95e4T^{2} \)
47 \( 1 + (89.8 - 155. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-218. - 378. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (286. + 496. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (368. + 212. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-728. + 420. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 1.17e3iT - 3.57e5T^{2} \)
73 \( 1 + (-716. + 413. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-282. - 162. i)T + (2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 507.T + 5.71e5T^{2} \)
89 \( 1 + (176. + 102. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.11e3iT - 9.12e5T^{2} \)
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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

−10.83211683569265927328190891247, −9.607361810493884810810848624568, −9.046626896794901301285252468877, −7.57822881980022749248942106901, −6.59745025856543775362442157801, −6.26694795829773359459319187491, −4.95144486173400651195138891766, −3.55729119740226090620351617935, −2.01330997001616826219917375973, −1.11055033503310064053121723260, 1.33726617737530191491384893122, 2.25369887337802688296658388721, 4.24274946909559392166941582175, 5.16453941407837586293930672524, 5.65215586786970183513928278797, 6.94265859108976052931682592236, 8.291074726339640865757636828081, 9.143052287967858577728112416318, 9.804103274756138252995321825677, 10.62114316271152389263581725899

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