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Properties

Label 2-448-112.27-c3-0-19
Degree $2$
Conductor $448$
Sign $0.999 + 0.0395i$
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  + (−6.75 + 6.75i)3-s + (−10.7 + 10.7i)5-s + (−7.90 + 16.7i)7-s − 64.1i·9-s + (−5.28 + 5.28i)11-s + (−50.4 − 50.4i)13-s − 145. i·15-s − 17.6i·17-s + (−63.5 + 63.5i)19-s + (−59.6 − 166. i)21-s − 7.51·23-s − 105. i·25-s + (250. + 250. i)27-s + (−153. + 153. i)29-s − 60.8·31-s + ⋯
L(s)  = 1  + (−1.29 + 1.29i)3-s + (−0.960 + 0.960i)5-s + (−0.427 + 0.904i)7-s − 2.37i·9-s + (−0.144 + 0.144i)11-s + (−1.07 − 1.07i)13-s − 2.49i·15-s − 0.252i·17-s + (−0.767 + 0.767i)19-s + (−0.619 − 1.72i)21-s − 0.0681·23-s − 0.846i·25-s + (1.78 + 1.78i)27-s + (−0.985 + 0.985i)29-s − 0.352·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0395i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0395i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.999 + 0.0395i$
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: \((2,\ 448,\ (\ :3/2),\ 0.999 + 0.0395i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.04137843485\)
\(L(\frac12)\) \(\approx\) \(0.04137843485\)
\(L(\frac{5}{2})\) 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 \( 1 + (7.90 - 16.7i)T \)
good3 \( 1 + (6.75 - 6.75i)T - 27iT^{2} \)
5 \( 1 + (10.7 - 10.7i)T - 125iT^{2} \)
11 \( 1 + (5.28 - 5.28i)T - 1.33e3iT^{2} \)
13 \( 1 + (50.4 + 50.4i)T + 2.19e3iT^{2} \)
17 \( 1 + 17.6iT - 4.91e3T^{2} \)
19 \( 1 + (63.5 - 63.5i)T - 6.85e3iT^{2} \)
23 \( 1 + 7.51T + 1.21e4T^{2} \)
29 \( 1 + (153. - 153. i)T - 2.43e4iT^{2} \)
31 \( 1 + 60.8T + 2.97e4T^{2} \)
37 \( 1 + (-129. - 129. i)T + 5.06e4iT^{2} \)
41 \( 1 - 48.6T + 6.89e4T^{2} \)
43 \( 1 + (123. - 123. i)T - 7.95e4iT^{2} \)
47 \( 1 + 254.T + 1.03e5T^{2} \)
53 \( 1 + (-498. - 498. i)T + 1.48e5iT^{2} \)
59 \( 1 + (-122. - 122. i)T + 2.05e5iT^{2} \)
61 \( 1 + (228. + 228. i)T + 2.26e5iT^{2} \)
67 \( 1 + (360. + 360. i)T + 3.00e5iT^{2} \)
71 \( 1 + 605.T + 3.57e5T^{2} \)
73 \( 1 - 913.T + 3.89e5T^{2} \)
79 \( 1 - 885. iT - 4.93e5T^{2} \)
83 \( 1 + (108. - 108. i)T - 5.71e5iT^{2} \)
89 \( 1 + 269.T + 7.04e5T^{2} \)
97 \( 1 - 1.45e3iT - 9.12e5T^{2} \)
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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

−10.66694686704496074244642312148, −10.07235899869138423419461088412, −9.197864232170518225608309127616, −7.84595461721996450741293141735, −6.74174274762740360651901113830, −5.77785185973644303040770963721, −4.98562495244843630009036247703, −3.82465526925826568247347325467, −2.90250903419116486877727288460, −0.03051687873364960910609745936, 0.57103944575874449199509264025, 1.98492641401225766786043181846, 4.15061823331898492087020137123, 4.92062912731103606876707471231, 6.12877036603204872036762556210, 7.11265519225913386185398051432, 7.55389789446307264360801178847, 8.638881895160060332357096722386, 9.948973927322909245990885981097, 11.12816303560503375323555774005

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