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Properties

Label 2-448-112.37-c1-0-0
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: $\chi_{448} (177, \cdot )$
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

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

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