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Properties

Label 2-448-28.27-c1-0-4
Degree $2$
Conductor $448$
Sign $0.912 + 0.409i$
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  − 2.61·3-s − 1.08i·5-s + (−1.08 + 2.41i)7-s + 3.82·9-s − 2i·11-s + 4.14i·13-s + 2.82i·15-s − 7.39i·17-s + 4.77·19-s + (2.82 − 6.30i)21-s − 3.65i·23-s + 3.82·25-s − 2.16·27-s + 7.65·29-s + 7.39·31-s + ⋯
L(s)  = 1  − 1.50·3-s − 0.484i·5-s + (−0.409 + 0.912i)7-s + 1.27·9-s − 0.603i·11-s + 1.14i·13-s + 0.730i·15-s − 1.79i·17-s + 1.09·19-s + (0.617 − 1.37i)21-s − 0.762i·23-s + 0.765·25-s − 0.416·27-s + 1.42·29-s + 1.32·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 + 0.409i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.912 + 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.912 + 0.409i$
Analytic conductor: \(3.57729\)
Root analytic conductor: \(1.89137\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (447, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1/2),\ 0.912 + 0.409i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.762317 - 0.163069i\)
\(L(\frac12)\) \(\approx\) \(0.762317 - 0.163069i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.08 - 2.41i)T \)
good3 \( 1 + 2.61T + 3T^{2} \)
5 \( 1 + 1.08iT - 5T^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
13 \( 1 - 4.14iT - 13T^{2} \)
17 \( 1 + 7.39iT - 17T^{2} \)
19 \( 1 - 4.77T + 19T^{2} \)
23 \( 1 + 3.65iT - 23T^{2} \)
29 \( 1 - 7.65T + 29T^{2} \)
31 \( 1 - 7.39T + 31T^{2} \)
37 \( 1 + 3.65T + 37T^{2} \)
41 \( 1 - 8.28iT - 41T^{2} \)
43 \( 1 + 7.65iT - 43T^{2} \)
47 \( 1 + 3.06T + 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 - 5.67T + 59T^{2} \)
61 \( 1 - 1.08iT - 61T^{2} \)
67 \( 1 + 4.34iT - 67T^{2} \)
71 \( 1 + 3.17iT - 71T^{2} \)
73 \( 1 + 0.896iT - 73T^{2} \)
79 \( 1 - 7.17iT - 79T^{2} \)
83 \( 1 - 1.71T + 83T^{2} \)
89 \( 1 + 5.22iT - 89T^{2} \)
97 \( 1 - 11.7iT - 97T^{2} \)
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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

−11.37998398336080397473806928599, −10.19844229290588962266369393512, −9.327793290859865552548289599274, −8.478727912845703248363424302080, −6.91991312350676617209888342529, −6.33126595591086199933063312437, −5.22297143646269708715168685827, −4.71553399971022081561718161508, −2.84766256422123525680131521515, −0.804770031749791529361631036948, 1.03761544070894097732922597542, 3.25814422942096498566976848587, 4.52151131842737184440575613934, 5.57446175637439549379913126684, 6.44994830582015943694621446871, 7.17911299187971880956736366289, 8.231480396678015622004648294336, 9.960274086155666893184491137046, 10.34749296341141763639729911292, 10.98913463037710649897581247657

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