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Properties

Label 2-448-56.3-c1-0-15
Degree $2$
Conductor $448$
Sign $-0.949 + 0.314i$
Analytic cond. $3.57729$
Root an. cond. $1.89137$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.182 + 0.105i)3-s + (−0.767 − 1.32i)5-s + (−2.19 − 1.47i)7-s + (−1.47 − 2.55i)9-s + (−2.37 + 4.11i)11-s − 4.95·13-s − 0.323i·15-s + (2.30 + 1.32i)17-s + (−3.74 + 2.16i)19-s + (−0.244 − 0.500i)21-s + (−0.547 + 0.316i)23-s + (1.32 − 2.29i)25-s − 1.25i·27-s − 7.85i·29-s + (−0.645 + 1.11i)31-s + ⋯
L(s)  = 1  + (0.105 + 0.0608i)3-s + (−0.343 − 0.594i)5-s + (−0.829 − 0.558i)7-s + (−0.492 − 0.853i)9-s + (−0.716 + 1.24i)11-s − 1.37·13-s − 0.0834i·15-s + (0.558 + 0.322i)17-s + (−0.858 + 0.495i)19-s + (−0.0534 − 0.109i)21-s + (−0.114 + 0.0659i)23-s + (0.264 − 0.458i)25-s − 0.241i·27-s − 1.45i·29-s + (−0.115 + 0.200i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 + 0.314i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.949 + 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.949 + 0.314i$
Analytic conductor: \(3.57729\)
Root analytic conductor: \(1.89137\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1/2),\ -0.949 + 0.314i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0579541 - 0.359678i\)
\(L(\frac12)\) \(\approx\) \(0.0579541 - 0.359678i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.19 + 1.47i)T \)
good3 \( 1 + (-0.182 - 0.105i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.767 + 1.32i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.37 - 4.11i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.95T + 13T^{2} \)
17 \( 1 + (-2.30 - 1.32i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.74 - 2.16i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.547 - 0.316i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.85iT - 29T^{2} \)
31 \( 1 + (0.645 - 1.11i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.5 - 0.866i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 7.85iT - 41T^{2} \)
43 \( 1 - 6.92T + 43T^{2} \)
47 \( 1 + (4.47 + 7.74i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.867 - 0.500i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (8.40 + 4.85i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.45 - 11.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.83 + 4.91i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6iT - 71T^{2} \)
73 \( 1 + (9.73 + 5.62i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.57 + 2.63i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 15.0iT - 83T^{2} \)
89 \( 1 + (4.5 - 2.59i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 6.96iT - 97T^{2} \)
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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

−10.33749126123681666019980393227, −9.931500974019562349508662346663, −8.976249785697853748510500353018, −7.87881964923341338196226933711, −7.09185601894235375242451194095, −5.98745081942247632164296289532, −4.73409050792902282135991877773, −3.81698253685945591136943707331, −2.41478689184453714148580560665, −0.20614985371845501427879996500, 2.60215337841826892284933947611, 3.17507769761620161452388353644, 4.93799438909503485443188554438, 5.81682250760372645287449806516, 6.95629853413040776035423063859, 7.84107244177546269663674893261, 8.741492595824572526455398492283, 9.732999286684917854482244185134, 10.76335770510280318876610558485, 11.28168774094930825804832491607

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