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Properties

Label 2-448-112.69-c2-0-12
Degree $2$
Conductor $448$
Sign $0.224 - 0.974i$
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  + (3.53 + 3.53i)3-s + (−0.208 + 0.208i)5-s + (1.63 − 6.80i)7-s + 15.9i·9-s + (−1.33 − 1.33i)11-s + (11.8 + 11.8i)13-s − 1.47·15-s + 15.6i·17-s + (13.1 + 13.1i)19-s + (29.8 − 18.2i)21-s + 14.3i·23-s + 24.9i·25-s + (−24.6 + 24.6i)27-s + (9.53 − 9.53i)29-s − 50.1i·31-s + ⋯
L(s)  = 1  + (1.17 + 1.17i)3-s + (−0.0417 + 0.0417i)5-s + (0.233 − 0.972i)7-s + 1.77i·9-s + (−0.121 − 0.121i)11-s + (0.911 + 0.911i)13-s − 0.0983·15-s + 0.922i·17-s + (0.690 + 0.690i)19-s + (1.42 − 0.870i)21-s + 0.622i·23-s + 0.996i·25-s + (−0.914 + 0.914i)27-s + (0.328 − 0.328i)29-s − 1.61i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.224 - 0.974i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.224 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.224 - 0.974i$
Analytic conductor: \(12.2071\)
Root analytic conductor: \(3.49386\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1),\ 0.224 - 0.974i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.655929415\)
\(L(\frac12)\) \(\approx\) \(2.655929415\)
\(L(2)\) 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 \( 1 + (-1.63 + 6.80i)T \)
good3 \( 1 + (-3.53 - 3.53i)T + 9iT^{2} \)
5 \( 1 + (0.208 - 0.208i)T - 25iT^{2} \)
11 \( 1 + (1.33 + 1.33i)T + 121iT^{2} \)
13 \( 1 + (-11.8 - 11.8i)T + 169iT^{2} \)
17 \( 1 - 15.6iT - 289T^{2} \)
19 \( 1 + (-13.1 - 13.1i)T + 361iT^{2} \)
23 \( 1 - 14.3iT - 529T^{2} \)
29 \( 1 + (-9.53 + 9.53i)T - 841iT^{2} \)
31 \( 1 + 50.1iT - 961T^{2} \)
37 \( 1 + (-38.9 - 38.9i)T + 1.36e3iT^{2} \)
41 \( 1 + 54.7T + 1.68e3T^{2} \)
43 \( 1 + (23.6 + 23.6i)T + 1.84e3iT^{2} \)
47 \( 1 + 8.49iT - 2.20e3T^{2} \)
53 \( 1 + (63.8 + 63.8i)T + 2.80e3iT^{2} \)
59 \( 1 + (-57.4 + 57.4i)T - 3.48e3iT^{2} \)
61 \( 1 + (40.2 + 40.2i)T + 3.72e3iT^{2} \)
67 \( 1 + (-9.59 + 9.59i)T - 4.48e3iT^{2} \)
71 \( 1 - 7.82iT - 5.04e3T^{2} \)
73 \( 1 - 86.7T + 5.32e3T^{2} \)
79 \( 1 - 56.6T + 6.24e3T^{2} \)
83 \( 1 + (64.2 + 64.2i)T + 6.88e3iT^{2} \)
89 \( 1 - 158.T + 7.92e3T^{2} \)
97 \( 1 - 136. iT - 9.40e3T^{2} \)
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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

−10.91665561499832305845724364140, −9.953722923582311435185877295652, −9.466199467499644490176048896270, −8.321418836167728818739960568387, −7.82428601240762205568255762776, −6.45262774888666232261237257550, −5.00082184870766086460616237197, −3.89227175279591187942021769533, −3.47284963141922077017864326374, −1.72764432285912057970309967401, 1.08651875917246339502549120245, 2.50053595095779540239562490167, 3.18462264805541690864694267358, 4.96866440705732152317370744278, 6.20117488436861559097370614496, 7.14791976121679285293134461287, 8.109679722355555881580750133102, 8.645909992792530944327244834982, 9.428731470205687996523993852357, 10.74283107055367637113792793077

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