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Properties

Label 2-448-7.2-c3-0-34
Degree $2$
Conductor $448$
Sign $-0.0425 + 0.999i$
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  + (0.130 + 0.226i)3-s + (9.75 − 16.8i)5-s + (−7.51 − 16.9i)7-s + (13.4 − 23.3i)9-s + (28.1 + 48.7i)11-s + 66.0·13-s + 5.10·15-s + (9.41 + 16.3i)17-s + (40.4 − 69.9i)19-s + (2.85 − 3.91i)21-s + (−44.6 + 77.3i)23-s + (−127. − 221. i)25-s + 14.1·27-s − 104.·29-s + (−74.2 − 128. i)31-s + ⋯
L(s)  = 1  + (0.0251 + 0.0435i)3-s + (0.872 − 1.51i)5-s + (−0.405 − 0.914i)7-s + (0.498 − 0.863i)9-s + (0.771 + 1.33i)11-s + 1.40·13-s + 0.0877·15-s + (0.134 + 0.232i)17-s + (0.487 − 0.845i)19-s + (0.0296 − 0.0406i)21-s + (−0.405 + 0.701i)23-s + (−1.02 − 1.76i)25-s + 0.100·27-s − 0.666·29-s + (−0.430 − 0.744i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0425 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0425 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.0425 + 0.999i$
Analytic conductor: \(26.4328\)
Root analytic conductor: \(5.14128\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :3/2),\ -0.0425 + 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.470993985\)
\(L(\frac12)\) \(\approx\) \(2.470993985\)
\(L(\frac{5}{2})\) 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 \( 1 + (7.51 + 16.9i)T \)
good3 \( 1 + (-0.130 - 0.226i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (-9.75 + 16.8i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-28.1 - 48.7i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 66.0T + 2.19e3T^{2} \)
17 \( 1 + (-9.41 - 16.3i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-40.4 + 69.9i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (44.6 - 77.3i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 104.T + 2.43e4T^{2} \)
31 \( 1 + (74.2 + 128. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-6.90 + 11.9i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 174.T + 6.89e4T^{2} \)
43 \( 1 - 205.T + 7.95e4T^{2} \)
47 \( 1 + (58.3 - 101. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (158. + 273. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-269. - 467. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-72.6 + 125. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (238. + 412. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 131.T + 3.57e5T^{2} \)
73 \( 1 + (174. + 302. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (403. - 698. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 233.T + 5.71e5T^{2} \)
89 \( 1 + (-267. + 463. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 80.7T + 9.12e5T^{2} \)
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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

−10.13823996594257405651755661960, −9.389322415470782521347180633392, −9.070151745893158946752137636135, −7.66643146227880847541966499512, −6.62938734922416376916007080038, −5.73631323396412417840500585442, −4.45185051197167053415025096703, −3.79952657187250814471360912087, −1.64478084759165372969438789774, −0.878236839187490819668073073503, 1.61420733395787571298718151117, 2.83027704761139693017030658908, 3.70762801442455127485315694133, 5.70241877504502366263330610825, 6.07093803195965310968162839024, 7.01586165893913500529852594301, 8.253380106162851329644057892527, 9.173274255986217784234823297784, 10.13745336871710890965554334427, 10.89724492900073472352301875982

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