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Properties

Label 2-448-112.19-c1-0-4
Degree $2$
Conductor $448$
Sign $0.884 - 0.466i$
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  + (−0.350 + 1.30i)3-s + (1.09 − 0.294i)5-s + (1.56 − 2.13i)7-s + (1.01 + 0.584i)9-s + (−0.259 + 0.969i)11-s + (0.430 − 0.430i)13-s + 1.53i·15-s + (3.08 − 1.78i)17-s + (−0.231 + 0.0621i)19-s + (2.23 + 2.79i)21-s + (3.02 − 5.23i)23-s + (−3.20 + 1.85i)25-s + (−3.98 + 3.98i)27-s + (4.74 + 4.74i)29-s + (3.96 + 6.86i)31-s + ⋯
L(s)  = 1  + (−0.202 + 0.754i)3-s + (0.491 − 0.131i)5-s + (0.592 − 0.805i)7-s + (0.337 + 0.194i)9-s + (−0.0783 + 0.292i)11-s + (0.119 − 0.119i)13-s + 0.397i·15-s + (0.748 − 0.432i)17-s + (−0.0531 + 0.0142i)19-s + (0.488 + 0.610i)21-s + (0.629 − 1.09i)23-s + (−0.641 + 0.370i)25-s + (−0.767 + 0.767i)27-s + (0.880 + 0.880i)29-s + (0.712 + 1.23i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.884 - 0.466i$
Analytic conductor: \(3.57729\)
Root analytic conductor: \(1.89137\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1/2),\ 0.884 - 0.466i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53684 + 0.380540i\)
\(L(\frac12)\) \(\approx\) \(1.53684 + 0.380540i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.56 + 2.13i)T \)
good3 \( 1 + (0.350 - 1.30i)T + (-2.59 - 1.5i)T^{2} \)
5 \( 1 + (-1.09 + 0.294i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.259 - 0.969i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-0.430 + 0.430i)T - 13iT^{2} \)
17 \( 1 + (-3.08 + 1.78i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.231 - 0.0621i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-3.02 + 5.23i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.74 - 4.74i)T + 29iT^{2} \)
31 \( 1 + (-3.96 - 6.86i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.75 - 6.53i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 6.90T + 41T^{2} \)
43 \( 1 + (-4.70 - 4.70i)T + 43iT^{2} \)
47 \( 1 + (-4.33 + 7.51i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.76 + 2.08i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (10.5 + 2.81i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (3.00 + 11.2i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-6.97 - 1.86i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 1.08T + 71T^{2} \)
73 \( 1 + (6.49 + 11.2i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.29 + 3.63i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.72 + 6.72i)T + 83iT^{2} \)
89 \( 1 + (-2.30 + 3.99i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 18.4iT - 97T^{2} \)
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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

−10.85254147547379235666476171489, −10.31774834900908777061671920224, −9.638614598504522766985738784553, −8.500326861832872339861213534306, −7.50120703779947641024591017890, −6.51006881560746432609514849361, −5.05949646468752559690744457095, −4.64253668345420510637073981206, −3.26172061192119534099987947513, −1.45043104777318262771259660290, 1.36497890319483515800668215421, 2.57961633060664630675786799853, 4.21012298487495245981981134440, 5.67151482244606728080905620913, 6.14975084644426476218889878724, 7.41258728549413691455599854670, 8.145081470375935194964735278234, 9.261590531966030787929177993994, 10.06850807876456414082288976175, 11.23339797482465294614245104186

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