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Properties

Label 2-448-28.3-c3-0-20
Degree $2$
Conductor $448$
Sign $0.490 - 0.871i$
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  + (−4.64 + 8.04i)3-s + (17.1 − 9.92i)5-s + (−18.3 − 2.13i)7-s + (−29.6 − 51.4i)9-s + (10.2 + 5.91i)11-s + 19.9i·13-s + 184. i·15-s + (1.57 + 0.908i)17-s + (3.66 + 6.34i)19-s + (102. − 138. i)21-s + (92.2 − 53.2i)23-s + (134. − 233. i)25-s + 300.·27-s + 191.·29-s + (62.5 − 108. i)31-s + ⋯
L(s)  = 1  + (−0.894 + 1.54i)3-s + (1.53 − 0.887i)5-s + (−0.993 − 0.115i)7-s + (−1.09 − 1.90i)9-s + (0.280 + 0.162i)11-s + 0.426i·13-s + 3.17i·15-s + (0.0224 + 0.0129i)17-s + (0.0442 + 0.0765i)19-s + (1.06 − 1.43i)21-s + (0.835 − 0.482i)23-s + (1.07 − 1.86i)25-s + 2.14·27-s + 1.22·29-s + (0.362 − 0.627i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.490 - 0.871i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.490 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.490 - 0.871i$
Analytic conductor: \(26.4328\)
Root analytic conductor: \(5.14128\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (255, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :3/2),\ 0.490 - 0.871i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.629098494\)
\(L(\frac12)\) \(\approx\) \(1.629098494\)
\(L(\frac{5}{2})\) 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 \( 1 + (18.3 + 2.13i)T \)
good3 \( 1 + (4.64 - 8.04i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (-17.1 + 9.92i)T + (62.5 - 108. i)T^{2} \)
11 \( 1 + (-10.2 - 5.91i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 19.9iT - 2.19e3T^{2} \)
17 \( 1 + (-1.57 - 0.908i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-3.66 - 6.34i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-92.2 + 53.2i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 191.T + 2.43e4T^{2} \)
31 \( 1 + (-62.5 + 108. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-158. - 274. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 321. iT - 6.89e4T^{2} \)
43 \( 1 - 74.3iT - 7.95e4T^{2} \)
47 \( 1 + (-77.2 - 133. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (159. - 276. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-30.5 + 52.9i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-267. + 154. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-514. - 297. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 48.6iT - 3.57e5T^{2} \)
73 \( 1 + (667. + 385. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-831. + 480. i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 1.06e3T + 5.71e5T^{2} \)
89 \( 1 + (567. - 327. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 704. iT - 9.12e5T^{2} \)
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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

−10.50555258498914494072615662174, −9.812028034536835551211124213489, −9.478290779603464221184626927809, −8.648277147957012557016749558346, −6.48424483978681993969082033807, −6.06857307944276772853227938914, −4.99178441908734743634873291268, −4.35991352752197713308002926062, −2.87679558991654803057034636555, −0.929313392384516992040831821713, 0.802018566437416210893770641774, 2.09200628111468260606438664574, 3.02225681362660782168733489126, 5.40995046895051254049890331506, 5.97909883854942806910482581879, 6.75307719884275887779280495186, 7.21377126854606517807982300458, 8.711386460318045274218107138211, 9.837240060376346754789826873626, 10.59833950075933939720038500687

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