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Properties

Label 2-448-448.227-c1-0-0
Degree $2$
Conductor $448$
Sign $-0.171 - 0.985i$
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  + (0.480 − 1.33i)2-s + (−0.903 + 1.03i)3-s + (−1.53 − 1.27i)4-s + (−0.0681 + 1.03i)5-s + (0.936 + 1.69i)6-s + (−1.01 − 2.44i)7-s + (−2.43 + 1.43i)8-s + (0.146 + 1.11i)9-s + (1.35 + 0.590i)10-s + (−0.907 + 2.67i)11-s + (2.70 − 0.429i)12-s + (−4.34 + 2.90i)13-s + (−3.73 + 0.169i)14-s + (−1.01 − 1.01i)15-s + (0.730 + 3.93i)16-s + (−0.941 − 3.51i)17-s + ⋯
L(s)  = 1  + (0.339 − 0.940i)2-s + (−0.521 + 0.595i)3-s + (−0.768 − 0.639i)4-s + (−0.0304 + 0.464i)5-s + (0.382 + 0.693i)6-s + (−0.382 − 0.924i)7-s + (−0.862 + 0.505i)8-s + (0.0487 + 0.370i)9-s + (0.426 + 0.186i)10-s + (−0.273 + 0.806i)11-s + (0.781 − 0.124i)12-s + (−1.20 + 0.805i)13-s + (−0.998 + 0.0452i)14-s + (−0.260 − 0.260i)15-s + (0.182 + 0.983i)16-s + (−0.228 − 0.852i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.171 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.171 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.171 - 0.985i$
Analytic conductor: \(3.57729\)
Root analytic conductor: \(1.89137\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1/2),\ -0.171 - 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.278839 + 0.331643i\)
\(L(\frac12)\) \(\approx\) \(0.278839 + 0.331643i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.480 + 1.33i)T \)
7 \( 1 + (1.01 + 2.44i)T \)
good3 \( 1 + (0.903 - 1.03i)T + (-0.391 - 2.97i)T^{2} \)
5 \( 1 + (0.0681 - 1.03i)T + (-4.95 - 0.652i)T^{2} \)
11 \( 1 + (0.907 - 2.67i)T + (-8.72 - 6.69i)T^{2} \)
13 \( 1 + (4.34 - 2.90i)T + (4.97 - 12.0i)T^{2} \)
17 \( 1 + (0.941 + 3.51i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (3.98 + 1.96i)T + (11.5 + 15.0i)T^{2} \)
23 \( 1 + (-0.977 - 7.42i)T + (-22.2 + 5.95i)T^{2} \)
29 \( 1 + (1.21 - 6.11i)T + (-26.7 - 11.0i)T^{2} \)
31 \( 1 + (3.68 + 2.12i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.680 - 10.3i)T + (-36.6 - 4.82i)T^{2} \)
41 \( 1 + (-3.12 - 1.29i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (1.82 + 9.18i)T + (-39.7 + 16.4i)T^{2} \)
47 \( 1 + (-0.886 + 3.30i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (10.0 + 3.41i)T + (42.0 + 32.2i)T^{2} \)
59 \( 1 + (-0.553 - 1.12i)T + (-35.9 + 46.8i)T^{2} \)
61 \( 1 + (-13.3 + 4.53i)T + (48.3 - 37.1i)T^{2} \)
67 \( 1 + (2.33 + 2.04i)T + (8.74 + 66.4i)T^{2} \)
71 \( 1 + (4.85 + 11.7i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (5.32 + 6.93i)T + (-18.8 + 70.5i)T^{2} \)
79 \( 1 + (-0.769 + 2.87i)T + (-68.4 - 39.5i)T^{2} \)
83 \( 1 + (-7.94 + 5.31i)T + (31.7 - 76.6i)T^{2} \)
89 \( 1 + (-5.78 - 4.43i)T + (23.0 + 85.9i)T^{2} \)
97 \( 1 + 12.7T + 97T^{2} \)
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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

−11.20001766464650555561603168729, −10.54965571112526948311161897605, −9.859521103718383555412203957854, −9.200519938137933277334061503797, −7.49055486767421312081584246243, −6.70258762488697619589153867139, −5.07044817282352926227295765327, −4.65378650554198549587892958958, −3.43254604912771197759846241334, −2.07250619585561396843552772383, 0.24541892457875803072677195436, 2.73283720294094615531486693153, 4.23928554500034043008777934995, 5.50386968313617777191760084923, 6.04013059775595542300281596521, 6.91686087867075856912791360298, 8.097675340605383985496385482206, 8.725010737111762232666266166856, 9.703011826594028306347866809984, 10.98683256573395225808365104826

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