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Properties

Label 2-448-7.5-c2-0-5
Degree $2$
Conductor $448$
Sign $-0.439 - 0.898i$
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  + (2.20 − 1.27i)3-s + (−3.56 − 2.05i)5-s + (−6.98 + 0.407i)7-s + (−1.27 + 2.20i)9-s + (1.63 + 2.83i)11-s + 5.88i·13-s − 10.4·15-s + (−12.0 + 6.93i)17-s + (13.7 + 7.91i)19-s + (−14.8 + 9.77i)21-s + (−18.2 + 31.6i)23-s + (−4.01 − 6.95i)25-s + 29.3i·27-s + 28.4·29-s + (−36.2 + 20.9i)31-s + ⋯
L(s)  = 1  + (0.733 − 0.423i)3-s + (−0.713 − 0.411i)5-s + (−0.998 + 0.0581i)7-s + (−0.141 + 0.244i)9-s + (0.148 + 0.257i)11-s + 0.452i·13-s − 0.697·15-s + (−0.707 + 0.408i)17-s + (0.721 + 0.416i)19-s + (−0.707 + 0.465i)21-s + (−0.793 + 1.37i)23-s + (−0.160 − 0.278i)25-s + 1.08i·27-s + 0.981·29-s + (−1.16 + 0.674i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.439 - 0.898i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.439 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.439 - 0.898i$
Analytic conductor: \(12.2071\)
Root analytic conductor: \(3.49386\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1),\ -0.439 - 0.898i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6636450725\)
\(L(\frac12)\) \(\approx\) \(0.6636450725\)
\(L(2)\) 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 \( 1 + (6.98 - 0.407i)T \)
good3 \( 1 + (-2.20 + 1.27i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (3.56 + 2.05i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-1.63 - 2.83i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 5.88iT - 169T^{2} \)
17 \( 1 + (12.0 - 6.93i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-13.7 - 7.91i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (18.2 - 31.6i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 28.4T + 841T^{2} \)
31 \( 1 + (36.2 - 20.9i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-7.14 + 12.3i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 21.3iT - 1.68e3T^{2} \)
43 \( 1 + 55.3T + 1.84e3T^{2} \)
47 \( 1 + (-29.3 - 16.9i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (42.4 + 73.4i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (58.5 - 33.7i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-25.6 - 14.7i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (27.4 + 47.5i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 83.8T + 5.04e3T^{2} \)
73 \( 1 + (108. - 62.8i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (35.1 - 60.9i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 27.1iT - 6.88e3T^{2} \)
89 \( 1 + (126. + 73.0i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 11.3iT - 9.40e3T^{2} \)
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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

−11.32459387023734257673772347865, −10.11451154853953185069224015754, −9.222816844613894390211005337248, −8.434843001043894232830176723290, −7.59699813605860911534012179916, −6.74630088978530749458369111036, −5.50676074045202381880658672796, −4.11499773214420562925569342948, −3.17982365005863738517319176298, −1.78268386409110953461732398244, 0.24068130871118399431717198436, 2.74033277816972000860178494347, 3.43555253705590987094239482598, 4.43575375974053101773540845172, 6.02260479991038668254113011956, 6.91184939646825401730573705037, 7.950687504491505699742473973693, 8.873601657319353269396396272372, 9.610744860410003848772280149199, 10.47683948581671584937239416918

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