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Properties

Label 2-448-112.19-c1-0-13
Degree $2$
Conductor $448$
Sign $-0.927 - 0.373i$
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  + (0.665 − 2.48i)3-s + (−3.12 + 0.837i)5-s + (−1.56 − 2.13i)7-s + (−3.13 − 1.80i)9-s + (−0.376 + 1.40i)11-s + (−3.11 + 3.11i)13-s + 8.31i·15-s + (2.02 − 1.16i)17-s + (−4.40 + 1.18i)19-s + (−6.34 + 2.45i)21-s + (1.15 − 1.99i)23-s + (4.72 − 2.73i)25-s + (−1.12 + 1.12i)27-s + (−1.55 − 1.55i)29-s + (−3.88 − 6.73i)31-s + ⋯
L(s)  = 1  + (0.384 − 1.43i)3-s + (−1.39 + 0.374i)5-s + (−0.590 − 0.807i)7-s + (−1.04 − 0.602i)9-s + (−0.113 + 0.423i)11-s + (−0.863 + 0.863i)13-s + 2.14i·15-s + (0.490 − 0.283i)17-s + (−1.01 + 0.270i)19-s + (−1.38 + 0.536i)21-s + (0.240 − 0.416i)23-s + (0.945 − 0.546i)25-s + (−0.216 + 0.216i)27-s + (−0.288 − 0.288i)29-s + (−0.698 − 1.20i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 - 0.373i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.927 - 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.927 - 0.373i$
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: \((2,\ 448,\ (\ :1/2),\ -0.927 - 0.373i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0803259 + 0.414527i\)
\(L(\frac12)\) \(\approx\) \(0.0803259 + 0.414527i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.665 + 2.48i)T + (-2.59 - 1.5i)T^{2} \)
5 \( 1 + (3.12 - 0.837i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.376 - 1.40i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (3.11 - 3.11i)T - 13iT^{2} \)
17 \( 1 + (-2.02 + 1.16i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.40 - 1.18i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-1.15 + 1.99i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.55 + 1.55i)T + 29iT^{2} \)
31 \( 1 + (3.88 + 6.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.272 - 1.01i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 2.77T + 41T^{2} \)
43 \( 1 + (7.12 + 7.12i)T + 43iT^{2} \)
47 \( 1 + (1.42 - 2.46i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-11.0 - 2.97i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (3.77 + 1.01i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (3.72 + 13.9i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (2.59 + 0.695i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 7.48T + 71T^{2} \)
73 \( 1 + (5.65 + 9.78i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.706 - 0.408i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.65 + 2.65i)T + 83iT^{2} \)
89 \( 1 + (2.40 - 4.17i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 5.86iT - 97T^{2} \)
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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

−10.75974879451346039553347993320, −9.658449082759752137631413670654, −8.415609813707422741717543105714, −7.52432068399576980876082865641, −7.18786666929448425897281250116, −6.37811109805106989261475756159, −4.49609084119022581310311235328, −3.46835486507065191959339141094, −2.12308148251220743799365354808, −0.24006533451651154365243938985, 2.96715074306237975754203021045, 3.67797766220875716717560347850, 4.74459390888821365599454511838, 5.58347944070559178104227365306, 7.17639119538536659417825159577, 8.370262961468286556414405267204, 8.781835807267608284618117744447, 9.836259236225720693393966594179, 10.57013531816546940079818015741, 11.51563421222398275964935475877

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