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Properties

Label 2-448-7.3-c2-0-10
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

\(L(\frac{3}{2})\) \(\approx\) \(1.984303901\)
\(L(\frac12)\) \(\approx\) \(1.984303901\)
\(L(2)\) 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

−11.13338130963706013132790806456, −9.969703026674376856676546957336, −9.382225134735112175135703472764, −8.155265783377731902339497631819, −7.67322942601452459371382462006, −6.44901398063113813030363657591, −5.12353264862754785970100621622, −4.20852085167624355916021796828, −3.02089886844481648966174950469, −1.58432754758495148290724390051, 0.845818008918471279983993120345, 2.52008020042059822408657505009, 3.54863778730955824575379814747, 5.21216995535126696558136508088, 5.63050702616825810646137342573, 7.52426191824515905999274677145, 7.974443942756281680692180286974, 8.522141782174098334965168724422, 9.886149592127492150780898862990, 10.79109388383812756510635553932

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