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Properties

Label 2-448-28.11-c0-0-1
Degree $2$
Conductor $448$
Sign $0.895 - 0.444i$
Analytic cond. $0.223581$
Root an. cond. $0.472843$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)3-s + (0.5 + 0.866i)5-s − i·7-s + (−0.866 − 0.5i)11-s + 0.999i·15-s + (−0.5 + 0.866i)17-s + (−0.866 + 0.5i)19-s + (0.5 − 0.866i)21-s + (−0.866 + 0.5i)23-s − i·27-s + (0.866 + 0.5i)31-s + (−0.499 − 0.866i)33-s + (0.866 − 0.5i)35-s + (−0.5 − 0.866i)37-s + (0.866 − 0.5i)47-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)3-s + (0.5 + 0.866i)5-s − i·7-s + (−0.866 − 0.5i)11-s + 0.999i·15-s + (−0.5 + 0.866i)17-s + (−0.866 + 0.5i)19-s + (0.5 − 0.866i)21-s + (−0.866 + 0.5i)23-s − i·27-s + (0.866 + 0.5i)31-s + (−0.499 − 0.866i)33-s + (0.866 − 0.5i)35-s + (−0.5 − 0.866i)37-s + (0.866 − 0.5i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.895 - 0.444i$
Analytic conductor: \(0.223581\)
Root analytic conductor: \(0.472843\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :0),\ 0.895 - 0.444i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.098456507\)
\(L(\frac12)\) \(\approx\) \(1.098456507\)
\(L(1)\) 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 \( 1 + iT \)
good3 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−10.91353623593695829153340298549, −10.45370560663271084131031910512, −9.786472057180894089644777048582, −8.602129690442198335375404039727, −7.909546975805198305303392438135, −6.73991759376615644985774005635, −5.86351708053610455691982706130, −4.26091385079697481830055125764, −3.40871186649369219385253896134, −2.27619036779853375285400128506, 2.02994065536684953209698623130, 2.73398387896804476451503107402, 4.64136410754007343002297009941, 5.44163912864693047746632767644, 6.65694898352044507335700182881, 7.87796513082127025455064011611, 8.564727706205373841052505983397, 9.197753547314721500721906544338, 10.13465358453283239459952849933, 11.37366971238623890730623490038

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