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Properties

Label 2-448-56.3-c1-0-7
Degree $2$
Conductor $448$
Sign $0.737 - 0.675i$
Analytic cond. $3.57729$
Root an. cond. $1.89137$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.48 + 1.43i)3-s + (0.241 + 0.418i)5-s + (0.447 − 2.60i)7-s + (2.60 + 4.51i)9-s + (−2.03 + 3.52i)11-s + 3.21·13-s + 1.38i·15-s + (−0.724 − 0.418i)17-s + (4.66 − 2.69i)19-s + (4.84 − 5.83i)21-s + (−7.44 + 4.29i)23-s + (2.38 − 4.12i)25-s + 6.34i·27-s − 4.36i·29-s + (−3.76 + 6.52i)31-s + ⋯
L(s)  = 1  + (1.43 + 0.827i)3-s + (0.107 + 0.186i)5-s + (0.169 − 0.985i)7-s + (0.869 + 1.50i)9-s + (−0.613 + 1.06i)11-s + 0.891·13-s + 0.357i·15-s + (−0.175 − 0.101i)17-s + (1.06 − 0.617i)19-s + (1.05 − 1.27i)21-s + (−1.55 + 0.896i)23-s + (0.476 − 0.825i)25-s + 1.22i·27-s − 0.809i·29-s + (−0.676 + 1.17i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.737 - 0.675i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.737 - 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.737 - 0.675i$
Analytic conductor: \(3.57729\)
Root analytic conductor: \(1.89137\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1/2),\ 0.737 - 0.675i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.09215 + 0.813880i\)
\(L(\frac12)\) \(\approx\) \(2.09215 + 0.813880i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.447 + 2.60i)T \)
good3 \( 1 + (-2.48 - 1.43i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.241 - 0.418i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.03 - 3.52i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.21T + 13T^{2} \)
17 \( 1 + (0.724 + 0.418i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.66 + 2.69i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (7.44 - 4.29i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.36iT - 29T^{2} \)
31 \( 1 + (3.76 - 6.52i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.5 - 0.866i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.36iT - 41T^{2} \)
43 \( 1 + 6.92T + 43T^{2} \)
47 \( 1 + (5.26 + 9.12i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-10.0 - 5.83i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.08 + 1.78i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.71 + 2.97i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.31 + 5.74i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6iT - 71T^{2} \)
73 \( 1 + (-5.54 - 3.20i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.58 + 0.915i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 11.0iT - 83T^{2} \)
89 \( 1 + (4.5 - 2.59i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 14.1iT - 97T^{2} \)
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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

−10.82164955106036974700803040880, −10.10598640885498093797221943778, −9.567182010305711542545434820086, −8.468875455269983510578648107396, −7.74405320641876633778210157537, −6.87016695357458975583186259284, −5.14493204478694819175417008663, −4.12312511088207275940391601909, −3.31759243099180418531045991029, −1.98265684878659700781353196098, 1.56775843491878276187955889822, 2.75092944160562869432950103816, 3.65836035094741668534305359077, 5.45327242548286827191221086700, 6.34366097956259865682973291571, 7.67586791294986247839545309036, 8.341722161132587536982904526806, 8.845505211924950944100014342703, 9.793914703369796267542525562054, 11.10188955630734686173952085888

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