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Properties

Label 2-448-7.3-c2-0-3
Degree $2$
Conductor $448$
Sign $0.386 - 0.922i$
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  + (−1.5 − 0.866i)3-s + (−1.5 + 0.866i)5-s − 7·7-s + (−3 − 5.19i)9-s + (7.5 − 12.9i)11-s + 13.8i·13-s + 3·15-s + (25.5 + 14.7i)17-s + (−13.5 + 7.79i)19-s + (10.5 + 6.06i)21-s + (4.5 + 7.79i)23-s + (−11 + 19.0i)25-s + 25.9i·27-s + 6·29-s + (−10.5 − 6.06i)31-s + ⋯
L(s)  = 1  + (−0.5 − 0.288i)3-s + (−0.300 + 0.173i)5-s − 7-s + (−0.333 − 0.577i)9-s + (0.681 − 1.18i)11-s + 1.06i·13-s + 0.200·15-s + (1.5 + 0.866i)17-s + (−0.710 + 0.410i)19-s + (0.5 + 0.288i)21-s + (0.195 + 0.338i)23-s + (−0.440 + 0.762i)25-s + 0.962i·27-s + 0.206·29-s + (−0.338 − 0.195i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(12.2071\)
Root analytic conductor: \(3.49386\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1),\ 0.386 - 0.922i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8459757379\)
\(L(\frac12)\) \(\approx\) \(0.8459757379\)
\(L(2)\) 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 \( 1 + 7T \)
good3 \( 1 + (1.5 + 0.866i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + (1.5 - 0.866i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-7.5 + 12.9i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 13.8iT - 169T^{2} \)
17 \( 1 + (-25.5 - 14.7i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (13.5 - 7.79i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-4.5 - 7.79i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 6T + 841T^{2} \)
31 \( 1 + (10.5 + 6.06i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-15.5 - 26.8i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 55.4iT - 1.68e3T^{2} \)
43 \( 1 + 10T + 1.84e3T^{2} \)
47 \( 1 + (-37.5 + 21.6i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (28.5 - 49.3i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-70.5 - 40.7i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-70.5 + 40.7i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (24.5 - 42.4i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 126T + 5.04e3T^{2} \)
73 \( 1 + (22.5 + 12.9i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-36.5 - 63.2i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 13.8iT - 6.88e3T^{2} \)
89 \( 1 + (-49.5 + 28.5i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 27.7iT - 9.40e3T^{2} \)
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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

−11.31016475505769084468000637574, −10.15163363363153894602868118615, −9.261237592090908129734438966881, −8.410498131781691505420745026852, −7.15886818748651979837878640564, −6.22867766418268130972660201851, −5.78558300361088415367323126453, −3.93181298136241106264934727785, −3.22454030322543449225101310417, −1.17547896419586736052545311069, 0.43569119496394259340297456663, 2.54080029384086394417128932754, 3.85366738851072586043550791785, 4.99159088387780405677823414530, 5.88450044165994735970789986146, 7.00961494906026401993462528382, 7.88298375083323534012116728480, 9.054360413879641639232750044749, 10.02039562547601080154939188489, 10.50747159561844844287385879075

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