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Properties

Label 2-448-112.69-c2-0-25
Degree $2$
Conductor $448$
Sign $-0.919 + 0.392i$
Analytic cond. $12.2071$
Root an. cond. $3.49386$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.00 − 1.00i)3-s + (4.03 − 4.03i)5-s + (−6.50 − 2.59i)7-s − 6.99i·9-s + (6.25 + 6.25i)11-s + (−4.97 − 4.97i)13-s − 8.08·15-s + 5.44i·17-s + (−17.3 − 17.3i)19-s + (3.91 + 9.10i)21-s + 8.07i·23-s − 7.62i·25-s + (−16.0 + 16.0i)27-s + (−27.3 + 27.3i)29-s − 53.1i·31-s + ⋯
L(s)  = 1  + (−0.333 − 0.333i)3-s + (0.807 − 0.807i)5-s + (−0.928 − 0.370i)7-s − 0.777i·9-s + (0.568 + 0.568i)11-s + (−0.382 − 0.382i)13-s − 0.539·15-s + 0.320i·17-s + (−0.911 − 0.911i)19-s + (0.186 + 0.433i)21-s + 0.351i·23-s − 0.305i·25-s + (−0.593 + 0.593i)27-s + (−0.943 + 0.943i)29-s − 1.71i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.919 + 0.392i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.919 + 0.392i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.919 + 0.392i$
Analytic conductor: \(12.2071\)
Root analytic conductor: \(3.49386\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1),\ -0.919 + 0.392i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9738835386\)
\(L(\frac12)\) \(\approx\) \(0.9738835386\)
\(L(2)\) 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 \( 1 + (6.50 + 2.59i)T \)
good3 \( 1 + (1.00 + 1.00i)T + 9iT^{2} \)
5 \( 1 + (-4.03 + 4.03i)T - 25iT^{2} \)
11 \( 1 + (-6.25 - 6.25i)T + 121iT^{2} \)
13 \( 1 + (4.97 + 4.97i)T + 169iT^{2} \)
17 \( 1 - 5.44iT - 289T^{2} \)
19 \( 1 + (17.3 + 17.3i)T + 361iT^{2} \)
23 \( 1 - 8.07iT - 529T^{2} \)
29 \( 1 + (27.3 - 27.3i)T - 841iT^{2} \)
31 \( 1 + 53.1iT - 961T^{2} \)
37 \( 1 + (-13.0 - 13.0i)T + 1.36e3iT^{2} \)
41 \( 1 - 3.54T + 1.68e3T^{2} \)
43 \( 1 + (13.9 + 13.9i)T + 1.84e3iT^{2} \)
47 \( 1 + 38.9iT - 2.20e3T^{2} \)
53 \( 1 + (51.8 + 51.8i)T + 2.80e3iT^{2} \)
59 \( 1 + (56.4 - 56.4i)T - 3.48e3iT^{2} \)
61 \( 1 + (84.3 + 84.3i)T + 3.72e3iT^{2} \)
67 \( 1 + (0.603 - 0.603i)T - 4.48e3iT^{2} \)
71 \( 1 + 78.2iT - 5.04e3T^{2} \)
73 \( 1 + 79.7T + 5.32e3T^{2} \)
79 \( 1 - 85.2T + 6.24e3T^{2} \)
83 \( 1 + (-5.96 - 5.96i)T + 6.88e3iT^{2} \)
89 \( 1 - 163.T + 7.92e3T^{2} \)
97 \( 1 - 89.6iT - 9.40e3T^{2} \)
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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

−10.35636039120590850106433270323, −9.364136295241965210932886894914, −9.130783616926109150049296598750, −7.56520910881885869491643939763, −6.57742448136239431056760355695, −5.93432449892660797797614105044, −4.73646742006774319054106715572, −3.49933673048141668051860944674, −1.83126216193808443791202079306, −0.40117155414364728735740425493, 2.05737147299026876580340810816, 3.17772818975491037658846642647, 4.54235879555331155898794948052, 5.92680708193478509550029000330, 6.29740980156355601940583032245, 7.47041974351548299595122533881, 8.773321364433195694926171040155, 9.669590567702417461203869584864, 10.36909014050302359882785276245, 11.05206224192616210940296882166

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