Properties

Label 2-448-7.6-c6-0-50
Degree $2$
Conductor $448$
Sign $-0.387 + 0.921i$
Analytic cond. $103.064$
Root an. cond. $10.1520$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 45.1i·3-s − 45.1i·5-s + (−133 + 316. i)7-s − 1.31e3·9-s + 874·11-s + 2.21e3i·13-s − 2.03e3·15-s + 5.96e3i·17-s − 3.11e3i·19-s + (1.42e4 + 6.00e3i)21-s − 4.73e3·23-s + 1.35e4·25-s + 2.62e4i·27-s − 1.11e4·29-s − 2.74e4i·31-s + ⋯
L(s)  = 1  − 1.67i·3-s − 0.361i·5-s + (−0.387 + 0.921i)7-s − 1.79·9-s + 0.656·11-s + 1.00i·13-s − 0.604·15-s + 1.21i·17-s − 0.454i·19-s + (1.54 + 0.648i)21-s − 0.389·23-s + 0.869·25-s + 1.33i·27-s − 0.457·29-s − 0.921i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.387 + 0.921i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.387 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.387 + 0.921i$
Analytic conductor: \(103.064\)
Root analytic conductor: \(10.1520\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :3),\ -0.387 + 0.921i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.764311609\)
\(L(\frac12)\) \(\approx\) \(1.764311609\)
\(L(4)\) 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 \( 1 + (133 - 316. i)T \)
good3 \( 1 + 45.1iT - 729T^{2} \)
5 \( 1 + 45.1iT - 1.56e4T^{2} \)
11 \( 1 - 874T + 1.77e6T^{2} \)
13 \( 1 - 2.21e3iT - 4.82e6T^{2} \)
17 \( 1 - 5.96e3iT - 2.41e7T^{2} \)
19 \( 1 + 3.11e3iT - 4.70e7T^{2} \)
23 \( 1 + 4.73e3T + 1.48e8T^{2} \)
29 \( 1 + 1.11e4T + 5.94e8T^{2} \)
31 \( 1 + 2.74e4iT - 8.87e8T^{2} \)
37 \( 1 + 3.00e3T + 2.56e9T^{2} \)
41 \( 1 + 5.75e4iT - 4.75e9T^{2} \)
43 \( 1 - 3.14e4T + 6.32e9T^{2} \)
47 \( 1 + 7.24e4iT - 1.07e10T^{2} \)
53 \( 1 - 7.64e4T + 2.21e10T^{2} \)
59 \( 1 + 1.13e5iT - 4.21e10T^{2} \)
61 \( 1 + 2.75e5iT - 5.15e10T^{2} \)
67 \( 1 - 4.95e5T + 9.04e10T^{2} \)
71 \( 1 - 1.84e5T + 1.28e11T^{2} \)
73 \( 1 - 6.09e4iT - 1.51e11T^{2} \)
79 \( 1 - 5.34e5T + 2.43e11T^{2} \)
83 \( 1 - 7.14e5iT - 3.26e11T^{2} \)
89 \( 1 - 6.29e5iT - 4.96e11T^{2} \)
97 \( 1 + 8.14e5iT - 8.32e11T^{2} \)
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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

−9.507041806854325750020528282384, −8.752542796544580267375474777308, −8.037963393590195261749465834626, −6.84029259910083236465689432262, −6.36869929724235442046223231743, −5.39629878087637645162000216612, −3.84012117886480853541166187784, −2.34868696604107579943828905698, −1.68589528426697196652929301677, −0.50505755600262888008954180074, 0.813408362395890664800792255971, 2.90547625683293012543094425557, 3.61103742007419907209339281895, 4.52127174721434531264762267451, 5.44375800127112965354859244971, 6.62348364268915526646110935035, 7.70249330048348284377377635346, 8.915033825210525443897666670966, 9.683069924513947417949015360203, 10.35975311234755960582448205832

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