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Properties

Label 2-448-28.27-c3-0-45
Degree $2$
Conductor $448$
Sign $-0.992 - 0.126i$
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  + 4.79·3-s − 17.0i·5-s + (−18.3 − 2.33i)7-s − 3.97·9-s − 41.4i·11-s + 45.3i·13-s − 81.9i·15-s + 28.2i·17-s − 41.5·19-s + (−88.1 − 11.2i)21-s + 93.9i·23-s − 166.·25-s − 148.·27-s − 27.8·29-s − 81.4·31-s + ⋯
L(s)  = 1  + 0.923·3-s − 1.52i·5-s + (−0.992 − 0.126i)7-s − 0.147·9-s − 1.13i·11-s + 0.967i·13-s − 1.41i·15-s + 0.403i·17-s − 0.501·19-s + (−0.916 − 0.116i)21-s + 0.851i·23-s − 1.33·25-s − 1.05·27-s − 0.178·29-s − 0.471·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.126i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.992 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.992 - 0.126i$
Analytic conductor: \(26.4328\)
Root analytic conductor: \(5.14128\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (447, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :3/2),\ -0.992 - 0.126i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7436797395\)
\(L(\frac12)\) \(\approx\) \(0.7436797395\)
\(L(\frac{5}{2})\) 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 \( 1 + (18.3 + 2.33i)T \)
good3 \( 1 - 4.79T + 27T^{2} \)
5 \( 1 + 17.0iT - 125T^{2} \)
11 \( 1 + 41.4iT - 1.33e3T^{2} \)
13 \( 1 - 45.3iT - 2.19e3T^{2} \)
17 \( 1 - 28.2iT - 4.91e3T^{2} \)
19 \( 1 + 41.5T + 6.85e3T^{2} \)
23 \( 1 - 93.9iT - 1.21e4T^{2} \)
29 \( 1 + 27.8T + 2.43e4T^{2} \)
31 \( 1 + 81.4T + 2.97e4T^{2} \)
37 \( 1 + 94.8T + 5.06e4T^{2} \)
41 \( 1 - 227. iT - 6.89e4T^{2} \)
43 \( 1 + 171. iT - 7.95e4T^{2} \)
47 \( 1 + 286.T + 1.03e5T^{2} \)
53 \( 1 - 575.T + 1.48e5T^{2} \)
59 \( 1 + 411.T + 2.05e5T^{2} \)
61 \( 1 + 778. iT - 2.26e5T^{2} \)
67 \( 1 + 198. iT - 3.00e5T^{2} \)
71 \( 1 - 197. iT - 3.57e5T^{2} \)
73 \( 1 + 255. iT - 3.89e5T^{2} \)
79 \( 1 + 1.17e3iT - 4.93e5T^{2} \)
83 \( 1 + 938.T + 5.71e5T^{2} \)
89 \( 1 + 1.16e3iT - 7.04e5T^{2} \)
97 \( 1 - 656. iT - 9.12e5T^{2} \)
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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

−9.857458375035815345968933298747, −8.993987045508317074928470167978, −8.727533046924291208726209570252, −7.74554010847430403118241796900, −6.37238362345141428048398446735, −5.42338778841472459174685779552, −4.10080820649845933072512788324, −3.23117352458711224387327225197, −1.71381069241821940253673964122, −0.19486779645350095541555745136, 2.38050755529281415935681797489, 2.93721112164131089305757529228, 3.93239377648149312450618886103, 5.65585583155490955480755610276, 6.75695894268566549523098021106, 7.34815765735841173566336127983, 8.417587186765092660546125122471, 9.478371626864496082059983594546, 10.18676318080714643183555247903, 10.88008165339678277814047933410

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