Properties

Label 2-448-7.4-c1-0-9
Degree $2$
Conductor $448$
Sign $-0.386 + 0.922i$
Analytic cond. $3.57729$
Root an. cond. $1.89137$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 + 2.59i)3-s + (−0.5 − 0.866i)5-s + (−2 + 1.73i)7-s + (−3 − 5.19i)9-s + (0.5 − 0.866i)11-s − 2·13-s + 3·15-s + (−1.5 + 2.59i)17-s + (−2.5 − 4.33i)19-s + (−1.5 − 7.79i)21-s + (−1.5 − 2.59i)23-s + (2 − 3.46i)25-s + 9·27-s + 6·29-s + (−0.5 + 0.866i)31-s + ⋯
L(s)  = 1  + (−0.866 + 1.49i)3-s + (−0.223 − 0.387i)5-s + (−0.755 + 0.654i)7-s + (−1 − 1.73i)9-s + (0.150 − 0.261i)11-s − 0.554·13-s + 0.774·15-s + (−0.363 + 0.630i)17-s + (−0.573 − 0.993i)19-s + (−0.327 − 1.70i)21-s + (−0.312 − 0.541i)23-s + (0.400 − 0.692i)25-s + 1.73·27-s + 1.11·29-s + (−0.0898 + 0.155i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.386 + 0.922i$
Analytic conductor: \(3.57729\)
Root analytic conductor: \(1.89137\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 448,\ (\ :1/2),\ -0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 - 1.73i)T \)
good3 \( 1 + (1.5 - 2.59i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.5 + 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.5 + 4.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.5 - 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.5 - 2.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.5 - 9.52i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 16T + 71T^{2} \)
73 \( 1 + (3.5 - 6.06i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.5 + 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 + (-4.5 - 7.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 6T + 97T^{2} \)
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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

−10.57740621618608805694527829361, −10.14626422500000299607201316635, −9.041946286245670320815761916392, −8.588684008479695694507575685081, −6.73422212067053265184060490740, −5.93401584933108306776253388564, −4.90056840148405923987728802608, −4.18055981698323905195836329405, −2.88487694441176343880457789316, 0, 1.67691603811137524108392664817, 3.21448565480825621746346424175, 4.83097929236300810904347161634, 6.08473424643705040383204548522, 6.84063791822644202189258750334, 7.33799546108765871606028141286, 8.340447768786625564727949574523, 9.815765458038132012689825511468, 10.61747997355707164025467188021

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