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Properties

Label 2-448-28.27-c1-0-12
Degree $2$
Conductor $448$
Sign $-0.156 + 0.987i$
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  + 1.08·3-s − 2.61i·5-s + (−2.61 − 0.414i)7-s − 1.82·9-s − 2i·11-s − 4.77i·13-s − 2.82i·15-s − 3.06i·17-s + 4.14·19-s + (−2.82 − 0.448i)21-s + 7.65i·23-s − 1.82·25-s − 5.22·27-s − 3.65·29-s + 3.06·31-s + ⋯
L(s)  = 1  + 0.624·3-s − 1.16i·5-s + (−0.987 − 0.156i)7-s − 0.609·9-s − 0.603i·11-s − 1.32i·13-s − 0.730i·15-s − 0.742i·17-s + 0.950·19-s + (−0.617 − 0.0978i)21-s + 1.59i·23-s − 0.365·25-s − 1.00·27-s − 0.679·29-s + 0.549·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.156 + 0.987i$
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: \((2,\ 448,\ (\ :1/2),\ -0.156 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.842600 - 0.986683i\)
\(L(\frac12)\) \(\approx\) \(0.842600 - 0.986683i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.61 + 0.414i)T \)
good3 \( 1 - 1.08T + 3T^{2} \)
5 \( 1 + 2.61iT - 5T^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
13 \( 1 + 4.77iT - 13T^{2} \)
17 \( 1 + 3.06iT - 17T^{2} \)
19 \( 1 - 4.14T + 19T^{2} \)
23 \( 1 - 7.65iT - 23T^{2} \)
29 \( 1 + 3.65T + 29T^{2} \)
31 \( 1 - 3.06T + 31T^{2} \)
37 \( 1 - 7.65T + 37T^{2} \)
41 \( 1 + 9.55iT - 41T^{2} \)
43 \( 1 - 3.65iT - 43T^{2} \)
47 \( 1 - 7.39T + 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 + 8.47T + 59T^{2} \)
61 \( 1 - 2.61iT - 61T^{2} \)
67 \( 1 + 15.6iT - 67T^{2} \)
71 \( 1 + 8.82iT - 71T^{2} \)
73 \( 1 - 12.6iT - 73T^{2} \)
79 \( 1 - 12.8iT - 79T^{2} \)
83 \( 1 - 11.5T + 83T^{2} \)
89 \( 1 - 2.16iT - 89T^{2} \)
97 \( 1 - 13.5iT - 97T^{2} \)
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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

−10.83472105073640759822076512684, −9.512179054834933854787713750848, −9.242507548792928406301079188210, −8.155283462305756720336753363887, −7.45443666582102853105860614775, −5.86689732241673862919584522221, −5.26630040715811996078485932876, −3.66575922407123753957752299589, −2.83958276893780968945561557379, −0.74907509829003426023875482046, 2.33545703404992749555995453362, 3.14551743724592692263013136129, 4.28646763947390039715300711441, 6.01583295969947298002820408569, 6.68851454492980175077468045331, 7.58781229597033722954586937016, 8.771078055427621756370998548163, 9.537016800705523471459787346045, 10.34221949533394055990980517351, 11.31146494712819115759820293304

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