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Properties

Label 2-448-112.27-c3-0-20
Degree $2$
Conductor $448$
Sign $0.781 - 0.623i$
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  + (3.57 − 3.57i)3-s + (2.98 − 2.98i)5-s + (−17.0 + 7.14i)7-s + 1.45i·9-s + (−10.4 + 10.4i)11-s + (53.8 + 53.8i)13-s − 21.3i·15-s − 29.4i·17-s + (−68.1 + 68.1i)19-s + (−35.5 + 86.6i)21-s + 108.·23-s + 107. i·25-s + (101. + 101. i)27-s + (−36.5 + 36.5i)29-s + 287.·31-s + ⋯
L(s)  = 1  + (0.687 − 0.687i)3-s + (0.267 − 0.267i)5-s + (−0.922 + 0.385i)7-s + 0.0538i·9-s + (−0.287 + 0.287i)11-s + (1.14 + 1.14i)13-s − 0.367i·15-s − 0.420i·17-s + (−0.822 + 0.822i)19-s + (−0.369 + 0.899i)21-s + 0.982·23-s + 0.857i·25-s + (0.724 + 0.724i)27-s + (−0.233 + 0.233i)29-s + 1.66·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.781 - 0.623i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.781 - 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.781 - 0.623i$
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: \((2,\ 448,\ (\ :3/2),\ 0.781 - 0.623i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.125734830\)
\(L(\frac12)\) \(\approx\) \(2.125734830\)
\(L(\frac{5}{2})\) 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 \( 1 + (17.0 - 7.14i)T \)
good3 \( 1 + (-3.57 + 3.57i)T - 27iT^{2} \)
5 \( 1 + (-2.98 + 2.98i)T - 125iT^{2} \)
11 \( 1 + (10.4 - 10.4i)T - 1.33e3iT^{2} \)
13 \( 1 + (-53.8 - 53.8i)T + 2.19e3iT^{2} \)
17 \( 1 + 29.4iT - 4.91e3T^{2} \)
19 \( 1 + (68.1 - 68.1i)T - 6.85e3iT^{2} \)
23 \( 1 - 108.T + 1.21e4T^{2} \)
29 \( 1 + (36.5 - 36.5i)T - 2.43e4iT^{2} \)
31 \( 1 - 287.T + 2.97e4T^{2} \)
37 \( 1 + (-38.5 - 38.5i)T + 5.06e4iT^{2} \)
41 \( 1 + 315.T + 6.89e4T^{2} \)
43 \( 1 + (-182. + 182. i)T - 7.95e4iT^{2} \)
47 \( 1 + 323.T + 1.03e5T^{2} \)
53 \( 1 + (8.29 + 8.29i)T + 1.48e5iT^{2} \)
59 \( 1 + (246. + 246. i)T + 2.05e5iT^{2} \)
61 \( 1 + (-289. - 289. i)T + 2.26e5iT^{2} \)
67 \( 1 + (-98.9 - 98.9i)T + 3.00e5iT^{2} \)
71 \( 1 - 1.13e3T + 3.57e5T^{2} \)
73 \( 1 + 653.T + 3.89e5T^{2} \)
79 \( 1 - 486. iT - 4.93e5T^{2} \)
83 \( 1 + (141. - 141. i)T - 5.71e5iT^{2} \)
89 \( 1 - 497.T + 7.04e5T^{2} \)
97 \( 1 - 1.55e3iT - 9.12e5T^{2} \)
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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

−10.74197123480206663558755826840, −9.670823537621374121776216011503, −8.869040935135564863334554027963, −8.206257684296474221979006545086, −6.98294978351606567266074300146, −6.33739918265368080884330564392, −5.07204778067798915629597999489, −3.65378164169492731276992909307, −2.48570065187673339042116768930, −1.39093068284234384183037432581, 0.66093501095708459165050692015, 2.77865050937312623964575092437, 3.42880971789184737438167811926, 4.53950460838578357897980774561, 6.03596301177167297319127422891, 6.67875008088306863128661055768, 8.150358340228916385479541664568, 8.771150391222301802350179626320, 9.808264674951821521486682486439, 10.39043438092803356682339045775

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