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Properties

Label 2-448-112.19-c1-0-12
Degree $2$
Conductor $448$
Sign $-0.569 + 0.821i$
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  + (0.825 − 3.08i)3-s + (1.86 − 0.501i)5-s + (−2.17 + 1.49i)7-s + (−6.21 − 3.58i)9-s + (0.944 − 3.52i)11-s + (0.372 − 0.372i)13-s − 6.17i·15-s + (2.39 − 1.38i)17-s + (−1.60 + 0.431i)19-s + (2.82 + 7.95i)21-s + (1.27 − 2.20i)23-s + (−1.08 + 0.626i)25-s + (−9.41 + 9.41i)27-s + (2.14 + 2.14i)29-s + (2.74 + 4.75i)31-s + ⋯
L(s)  = 1  + (0.476 − 1.77i)3-s + (0.836 − 0.224i)5-s + (−0.823 + 0.566i)7-s + (−2.07 − 1.19i)9-s + (0.284 − 1.06i)11-s + (0.103 − 0.103i)13-s − 1.59i·15-s + (0.582 − 0.336i)17-s + (−0.369 + 0.0989i)19-s + (0.615 + 1.73i)21-s + (0.265 − 0.459i)23-s + (−0.216 + 0.125i)25-s + (−1.81 + 1.81i)27-s + (0.397 + 0.397i)29-s + (0.493 + 0.854i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.569 + 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.569 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.569 + 0.821i$
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: \((2,\ 448,\ (\ :1/2),\ -0.569 + 0.821i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.749608 - 1.43140i\)
\(L(\frac12)\) \(\approx\) \(0.749608 - 1.43140i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.17 - 1.49i)T \)
good3 \( 1 + (-0.825 + 3.08i)T + (-2.59 - 1.5i)T^{2} \)
5 \( 1 + (-1.86 + 0.501i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.944 + 3.52i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-0.372 + 0.372i)T - 13iT^{2} \)
17 \( 1 + (-2.39 + 1.38i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.60 - 0.431i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-1.27 + 2.20i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.14 - 2.14i)T + 29iT^{2} \)
31 \( 1 + (-2.74 - 4.75i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.21 + 4.53i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 3.95T + 41T^{2} \)
43 \( 1 + (-4.29 - 4.29i)T + 43iT^{2} \)
47 \( 1 + (1.85 - 3.21i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.49 - 1.74i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (8.21 + 2.20i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.101 - 0.378i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-13.8 - 3.72i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 8.98T + 71T^{2} \)
73 \( 1 + (-0.766 - 1.32i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.47 + 2.00i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.67 - 3.67i)T + 83iT^{2} \)
89 \( 1 + (-3.35 + 5.80i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 4.52iT - 97T^{2} \)
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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

−10.94806187644290074630371056740, −9.568535039546818403242078005634, −8.832141828479618833179332278346, −8.109070403016293250485054829846, −6.93995218844565540074400126468, −6.19596576522618340101671595234, −5.57308652822555918215716871773, −3.27925773316531355267714158729, −2.37208234623799393794837371889, −1.00309516965368094421155969602, 2.42853122062574701588846221972, 3.64297194469878309189037133644, 4.42701267047640422797176173840, 5.56868580727584080192673432194, 6.60661807806550390834578723292, 7.962748295670356628139893769386, 9.174118892927537380343956129656, 9.779213624448120447901704480633, 10.15526362033631780053828821098, 10.97236702384040227315610871419

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