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Properties

Label 2-448-28.3-c3-0-43
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

\(L(2)\) \(\approx\) \(3.734811437\)
\(L(\frac12)\) \(\approx\) \(3.734811437\)
\(L(\frac{5}{2})\) 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

−10.15775319475558745972831898355, −9.174206122705192078286444910414, −8.482367423828915532132306956784, −7.84643725060030854895866498016, −6.62838936287070138044239159241, −5.83841683327470571752761105536, −4.72930687488766897095954039845, −2.75336740594098994772612126508, −1.78103710856715080509525701852, −1.17109506126344612125174283248, 2.05994224591930848417137382340, 2.83020449246733956254966384498, 4.12828127765004531441476179961, 5.15655949628858412992231049661, 6.00138188649585707213959942940, 7.50951005330491761432686653016, 8.519219632433237251724886452558, 9.351694890073884548748896656261, 10.29620605211640055371109137158, 10.41201531931522673104724601498

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