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Properties

Label 2-448-112.109-c1-0-12
Degree $2$
Conductor $448$
Sign $-0.940 + 0.339i$
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  + (−1.16 + 0.312i)3-s + (0.980 + 0.262i)5-s + (−2.60 + 0.476i)7-s + (−1.33 + 0.772i)9-s + (−0.635 − 2.36i)11-s + (−2.65 − 2.65i)13-s − 1.22·15-s + (−0.509 + 0.881i)17-s + (0.0250 − 0.0936i)19-s + (2.88 − 1.36i)21-s + (−1.67 + 0.965i)23-s + (−3.43 − 1.98i)25-s + (3.87 − 3.87i)27-s + (−5.05 − 5.05i)29-s + (−4.28 + 7.41i)31-s + ⋯
L(s)  = 1  + (−0.672 + 0.180i)3-s + (0.438 + 0.117i)5-s + (−0.983 + 0.180i)7-s + (−0.445 + 0.257i)9-s + (−0.191 − 0.714i)11-s + (−0.737 − 0.737i)13-s − 0.316·15-s + (−0.123 + 0.213i)17-s + (0.00575 − 0.0214i)19-s + (0.629 − 0.298i)21-s + (−0.348 + 0.201i)23-s + (−0.687 − 0.396i)25-s + (0.746 − 0.746i)27-s + (−0.938 − 0.938i)29-s + (−0.769 + 1.33i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.940 + 0.339i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.940 + 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.940 + 0.339i$
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: \((2,\ 448,\ (\ :1/2),\ -0.940 + 0.339i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0141992 - 0.0812417i\)
\(L(\frac12)\) \(\approx\) \(0.0141992 - 0.0812417i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.60 - 0.476i)T \)
good3 \( 1 + (1.16 - 0.312i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (-0.980 - 0.262i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.635 + 2.36i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (2.65 + 2.65i)T + 13iT^{2} \)
17 \( 1 + (0.509 - 0.881i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.0250 + 0.0936i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (1.67 - 0.965i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.05 + 5.05i)T + 29iT^{2} \)
31 \( 1 + (4.28 - 7.41i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (7.71 + 2.06i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 8.51iT - 41T^{2} \)
43 \( 1 + (-4.47 + 4.47i)T - 43iT^{2} \)
47 \( 1 + (6.02 + 10.4i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.381 - 1.42i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-1.86 - 6.96i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (1.18 - 4.42i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-3.38 + 0.907i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 5.43iT - 71T^{2} \)
73 \( 1 + (-7.34 - 4.23i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.433 - 0.751i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.44 - 5.44i)T + 83iT^{2} \)
89 \( 1 + (-3.93 + 2.26i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 16.4T + 97T^{2} \)
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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

−10.55310357113560121191545119516, −10.03917513619991080833346862208, −9.001746849548512973279248568267, −7.974378673799626301579200097109, −6.75341005779759768340692869967, −5.79662268114856462117608433818, −5.28271362409687788607320008363, −3.63350931199238826498188034126, −2.44816685699041914315040147179, −0.05103853881081624948931973705, 2.08568581229647614551470659521, 3.59005599376938054809235686475, 4.96619311908497177552771317050, 5.92243039009959224581035123405, 6.75418972615967909810964672480, 7.58937541488533849004119946907, 9.190397016047871728610991599687, 9.557568732702986677245027007606, 10.62607011346766831940183845000, 11.54676137423553737594921460702

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