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Properties

Label 2-448-7.3-c2-0-11
Degree $2$
Conductor $448$
Sign $-0.108 - 0.994i$
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  + (4.19 + 2.41i)3-s + (0.0446 − 0.0257i)5-s + (−6.12 + 3.39i)7-s + (7.20 + 12.4i)9-s + (0.894 − 1.55i)11-s + 5.87i·13-s + 0.249·15-s + (23.0 + 13.2i)17-s + (−22.8 + 13.1i)19-s + (−33.8 − 0.603i)21-s + (12.8 + 22.2i)23-s + (−12.4 + 21.6i)25-s + 26.1i·27-s + 27.1·29-s + (25.7 + 14.8i)31-s + ⋯
L(s)  = 1  + (1.39 + 0.806i)3-s + (0.00892 − 0.00515i)5-s + (−0.874 + 0.484i)7-s + (0.800 + 1.38i)9-s + (0.0813 − 0.140i)11-s + 0.452i·13-s + 0.0166·15-s + (1.35 + 0.781i)17-s + (−1.20 + 0.693i)19-s + (−1.61 − 0.0287i)21-s + (0.558 + 0.966i)23-s + (−0.499 + 0.865i)25-s + 0.969i·27-s + 0.937·29-s + (0.829 + 0.479i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.108 - 0.994i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.108 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.108 - 0.994i$
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: \((2,\ 448,\ (\ :1),\ -0.108 - 0.994i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.426937167\)
\(L(\frac12)\) \(\approx\) \(2.426937167\)
\(L(2)\) 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 \( 1 + (6.12 - 3.39i)T \)
good3 \( 1 + (-4.19 - 2.41i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + (-0.0446 + 0.0257i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-0.894 + 1.55i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 5.87iT - 169T^{2} \)
17 \( 1 + (-23.0 - 13.2i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (22.8 - 13.1i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-12.8 - 22.2i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 27.1T + 841T^{2} \)
31 \( 1 + (-25.7 - 14.8i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (30.8 + 53.4i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 65.7iT - 1.68e3T^{2} \)
43 \( 1 - 9.52T + 1.84e3T^{2} \)
47 \( 1 + (61.2 - 35.3i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-4.86 + 8.42i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (54.3 + 31.3i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-66.1 + 38.2i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-51.5 + 89.2i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 90.1T + 5.04e3T^{2} \)
73 \( 1 + (28.8 + 16.6i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-32.4 - 56.1i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 29.1iT - 6.88e3T^{2} \)
89 \( 1 + (-18.7 + 10.8i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 123. iT - 9.40e3T^{2} \)
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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

−10.79829939298463079505202693712, −9.962456558032769511245955343540, −9.337214158706509179109728121522, −8.573606768062814144458925608726, −7.77794322620527656488619928662, −6.47975152073605162562090110791, −5.30920437332136523627403496308, −3.83086412826279568769813807141, −3.32498930554414213419116514242, −1.99420726113581613479973195956, 0.877853999538807751814789406987, 2.56588597794080172205858230799, 3.25011307997176041162235754106, 4.59679039290846674820343317838, 6.36657143223568278182937394092, 6.98696621750532507345278045200, 8.036815467927181988718510868182, 8.598546831595295398773839936355, 9.752456638701695539638036859323, 10.27066099094066069774529255452

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