Properties

Label 2-448-112.109-c1-0-6
Degree $2$
Conductor $448$
Sign $0.499 + 0.866i$
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  + (−1.95 + 0.523i)3-s + (0.959 + 0.256i)5-s + (−0.292 − 2.62i)7-s + (0.941 − 0.543i)9-s + (0.505 + 1.88i)11-s + (−2.10 − 2.10i)13-s − 2.00·15-s + (2.83 − 4.91i)17-s + (0.165 − 0.616i)19-s + (1.94 + 4.98i)21-s + (5.92 − 3.42i)23-s + (−3.47 − 2.00i)25-s + (2.73 − 2.73i)27-s + (−0.207 − 0.207i)29-s + (3.94 − 6.83i)31-s + ⋯
L(s)  = 1  + (−1.12 + 0.302i)3-s + (0.428 + 0.114i)5-s + (−0.110 − 0.993i)7-s + (0.313 − 0.181i)9-s + (0.152 + 0.569i)11-s + (−0.583 − 0.583i)13-s − 0.518·15-s + (0.688 − 1.19i)17-s + (0.0379 − 0.141i)19-s + (0.425 + 1.08i)21-s + (1.23 − 0.713i)23-s + (−0.695 − 0.401i)25-s + (0.526 − 0.526i)27-s + (−0.0384 − 0.0384i)29-s + (0.708 − 1.22i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.499 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.499 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.499 + 0.866i$
Analytic conductor: \(3.57729\)
Root analytic conductor: \(1.89137\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1/2),\ 0.499 + 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.750276 - 0.433687i\)
\(L(\frac12)\) \(\approx\) \(0.750276 - 0.433687i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.292 + 2.62i)T \)
good3 \( 1 + (1.95 - 0.523i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (-0.959 - 0.256i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.505 - 1.88i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (2.10 + 2.10i)T + 13iT^{2} \)
17 \( 1 + (-2.83 + 4.91i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.165 + 0.616i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-5.92 + 3.42i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.207 + 0.207i)T + 29iT^{2} \)
31 \( 1 + (-3.94 + 6.83i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-9.60 - 2.57i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 2.40iT - 41T^{2} \)
43 \( 1 + (3.65 - 3.65i)T - 43iT^{2} \)
47 \( 1 + (-0.144 - 0.250i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.04 + 7.62i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (3.60 + 13.4i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.969 - 3.61i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (10.0 - 2.69i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 11.5iT - 71T^{2} \)
73 \( 1 + (0.310 + 0.179i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.84 - 6.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.424 - 0.424i)T + 83iT^{2} \)
89 \( 1 + (15.2 - 8.79i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 12.2T + 97T^{2} \)
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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

−11.01016598974507945159763097831, −9.923646068383060056015689274439, −9.747909442972234729153775908353, −8.023525531615350638577184825021, −7.08861537168506551447449592365, −6.21024963941346251450815503740, −5.13836058364277930605572884267, −4.41383761035643363663506014982, −2.77262729633410364016877452627, −0.67066227405119755444041067425, 1.48047641409919899169552739656, 3.12376543602310466320191899225, 4.82237768965681609782626005566, 5.80897153911305589114216687615, 6.19161181451536878366642875766, 7.41598982545176417618210756860, 8.675847401453261928390404791472, 9.436065282966905942782254457619, 10.50929357843500871035884083994, 11.40731728419257254287743621274

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