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Properties

Label 2-448-448.195-c1-0-47
Degree $2$
Conductor $448$
Sign $-0.946 + 0.321i$
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  + (−0.771 − 1.18i)2-s + (0.440 − 2.21i)3-s + (−0.809 + 1.82i)4-s + (0.656 − 0.981i)5-s + (−2.96 + 1.18i)6-s + (2.05 + 1.66i)7-s + (2.79 − 0.451i)8-s + (−1.93 − 0.800i)9-s + (−1.66 − 0.0200i)10-s + (−0.644 − 3.23i)11-s + (3.69 + 2.59i)12-s + (−3.09 − 4.63i)13-s + (0.385 − 3.72i)14-s + (−1.88 − 1.88i)15-s + (−2.68 − 2.96i)16-s + (2.13 − 2.13i)17-s + ⋯
L(s)  = 1  + (−0.545 − 0.838i)2-s + (0.254 − 1.27i)3-s + (−0.404 + 0.914i)4-s + (0.293 − 0.439i)5-s + (−1.20 + 0.484i)6-s + (0.777 + 0.628i)7-s + (0.987 − 0.159i)8-s + (−0.643 − 0.266i)9-s + (−0.528 − 0.00635i)10-s + (−0.194 − 0.976i)11-s + (1.06 + 0.749i)12-s + (−0.858 − 1.28i)13-s + (0.102 − 0.994i)14-s + (−0.486 − 0.486i)15-s + (−0.672 − 0.740i)16-s + (0.518 − 0.518i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.946 + 0.321i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.946 + 0.321i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.946 + 0.321i$
Analytic conductor: \(3.57729\)
Root analytic conductor: \(1.89137\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (195, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1/2),\ -0.946 + 0.321i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.189586 - 1.14702i\)
\(L(\frac12)\) \(\approx\) \(0.189586 - 1.14702i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.771 + 1.18i)T \)
7 \( 1 + (-2.05 - 1.66i)T \)
good3 \( 1 + (-0.440 + 2.21i)T + (-2.77 - 1.14i)T^{2} \)
5 \( 1 + (-0.656 + 0.981i)T + (-1.91 - 4.61i)T^{2} \)
11 \( 1 + (0.644 + 3.23i)T + (-10.1 + 4.20i)T^{2} \)
13 \( 1 + (3.09 + 4.63i)T + (-4.97 + 12.0i)T^{2} \)
17 \( 1 + (-2.13 + 2.13i)T - 17iT^{2} \)
19 \( 1 + (-1.46 - 2.19i)T + (-7.27 + 17.5i)T^{2} \)
23 \( 1 + (8.43 + 3.49i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (-1.04 - 0.208i)T + (26.7 + 11.0i)T^{2} \)
31 \( 1 - 4.18iT - 31T^{2} \)
37 \( 1 + (0.883 - 1.32i)T + (-14.1 - 34.1i)T^{2} \)
41 \( 1 + (-11.2 - 4.64i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (1.78 - 0.355i)T + (39.7 - 16.4i)T^{2} \)
47 \( 1 + (2.46 + 2.46i)T + 47iT^{2} \)
53 \( 1 + (-12.3 + 2.44i)T + (48.9 - 20.2i)T^{2} \)
59 \( 1 + (8.23 + 5.50i)T + (22.5 + 54.5i)T^{2} \)
61 \( 1 + (11.2 + 2.23i)T + (56.3 + 23.3i)T^{2} \)
67 \( 1 + (2.06 + 0.410i)T + (61.8 + 25.6i)T^{2} \)
71 \( 1 + (-4.26 - 10.2i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (-4.36 + 10.5i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (2.27 + 2.27i)T + 79iT^{2} \)
83 \( 1 + (-2.04 - 3.06i)T + (-31.7 + 76.6i)T^{2} \)
89 \( 1 + (2.17 - 0.899i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 - 7.53T + 97T^{2} \)
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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

−10.72739730181214432179259175974, −9.818840061906649617992636042003, −8.694268750116478423123421100116, −8.007761745708939176238886694144, −7.54596296281324815804338504071, −5.95940073437758546268606114095, −4.92018843716278504651015534788, −3.10094632670024067642134834206, −2.10772702100249261560307756639, −0.892896812914886136488303537335, 2.02486303279133188557047490563, 4.17548493635495098707111182913, 4.60754481317953512623624431285, 5.83589276668761788611815414628, 7.12545115917080530793566972195, 7.75405619944324356067356312080, 8.982065336880322643873163451078, 9.796146719840554965957347272872, 10.18027511957562969060358777334, 11.03299749046034660273675212997

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