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Properties

Label 2-448-112.27-c3-0-13
Degree $2$
Conductor $448$
Sign $-0.612 - 0.790i$
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  + (−7.12 + 7.12i)3-s + (5.44 − 5.44i)5-s + (−16.6 − 8.05i)7-s − 74.5i·9-s + (12.2 − 12.2i)11-s + (14.9 + 14.9i)13-s + 77.5i·15-s + 97.2i·17-s + (65.2 − 65.2i)19-s + (176. − 61.4i)21-s − 51.7·23-s + 65.7i·25-s + (338. + 338. i)27-s + (19.0 − 19.0i)29-s + 112.·31-s + ⋯
L(s)  = 1  + (−1.37 + 1.37i)3-s + (0.486 − 0.486i)5-s + (−0.900 − 0.435i)7-s − 2.76i·9-s + (0.336 − 0.336i)11-s + (0.318 + 0.318i)13-s + 1.33i·15-s + 1.38i·17-s + (0.787 − 0.787i)19-s + (1.83 − 0.638i)21-s − 0.469·23-s + 0.525i·25-s + (2.41 + 2.41i)27-s + (0.121 − 0.121i)29-s + 0.652·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.612 - 0.790i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.612 - 0.790i$
Analytic conductor: \(26.4328\)
Root analytic conductor: \(5.14128\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (335, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :3/2),\ -0.612 - 0.790i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7234019424\)
\(L(\frac12)\) \(\approx\) \(0.7234019424\)
\(L(\frac{5}{2})\) 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 \( 1 + (16.6 + 8.05i)T \)
good3 \( 1 + (7.12 - 7.12i)T - 27iT^{2} \)
5 \( 1 + (-5.44 + 5.44i)T - 125iT^{2} \)
11 \( 1 + (-12.2 + 12.2i)T - 1.33e3iT^{2} \)
13 \( 1 + (-14.9 - 14.9i)T + 2.19e3iT^{2} \)
17 \( 1 - 97.2iT - 4.91e3T^{2} \)
19 \( 1 + (-65.2 + 65.2i)T - 6.85e3iT^{2} \)
23 \( 1 + 51.7T + 1.21e4T^{2} \)
29 \( 1 + (-19.0 + 19.0i)T - 2.43e4iT^{2} \)
31 \( 1 - 112.T + 2.97e4T^{2} \)
37 \( 1 + (292. + 292. i)T + 5.06e4iT^{2} \)
41 \( 1 - 145.T + 6.89e4T^{2} \)
43 \( 1 + (180. - 180. i)T - 7.95e4iT^{2} \)
47 \( 1 + 325.T + 1.03e5T^{2} \)
53 \( 1 + (-19.7 - 19.7i)T + 1.48e5iT^{2} \)
59 \( 1 + (-51.1 - 51.1i)T + 2.05e5iT^{2} \)
61 \( 1 + (-286. - 286. i)T + 2.26e5iT^{2} \)
67 \( 1 + (-529. - 529. i)T + 3.00e5iT^{2} \)
71 \( 1 + 246.T + 3.57e5T^{2} \)
73 \( 1 + 798.T + 3.89e5T^{2} \)
79 \( 1 - 629. iT - 4.93e5T^{2} \)
83 \( 1 + (170. - 170. i)T - 5.71e5iT^{2} \)
89 \( 1 - 113.T + 7.04e5T^{2} \)
97 \( 1 - 1.41e3iT - 9.12e5T^{2} \)
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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

−10.89127745681539047375876176829, −10.11188692772097539066670059378, −9.500881333405775134728927196328, −8.720742016761683834710391541340, −6.85896485958697704121138871607, −6.06814783566427568584995632743, −5.36559205481661042697520023892, −4.23531900685831581931361338173, −3.47307255290809938020883620801, −1.01227218767807607058096187438, 0.34856389817251497167561884207, 1.72403387288010833285413668987, 2.96729031132632836308463206374, 4.99981223093447614656082634807, 5.88187638688759557265329199786, 6.57154079237715100650302773204, 7.18604099843084416213902652012, 8.294240438703296623326382130840, 9.791125458733655099280875369264, 10.39036482886498167926141701477

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