↑

Properties

Label 2-448-7.4-c1-0-0
Degree $2$
Conductor $448$
Sign $-0.922 - 0.386i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 + 1.5i)3-s + (0.5 + 0.866i)5-s + (−1.73 − 2i)7-s + (−2.59 + 4.5i)11-s − 1.73·15-s + (−2.5 + 4.33i)17-s + (−0.866 − 1.5i)19-s + (4.5 − 0.866i)21-s + (−0.866 − 1.5i)23-s + (2 − 3.46i)25-s − 5.19·27-s − 8·29-s + (−4.33 + 7.5i)31-s + (−4.5 − 7.79i)33-s + (0.866 − 2.5i)35-s + ⋯
L(s)  = 1  + (−0.499 + 0.866i)3-s + (0.223 + 0.387i)5-s + (−0.654 − 0.755i)7-s + (−0.783 + 1.35i)11-s − 0.447·15-s + (−0.606 + 1.05i)17-s + (−0.198 − 0.344i)19-s + (0.981 − 0.188i)21-s + (−0.180 − 0.312i)23-s + (0.400 − 0.692i)25-s − 1.00·27-s − 1.48·29-s + (−0.777 + 1.34i)31-s + (−0.783 − 1.35i)33-s + (0.146 − 0.422i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.922 - 0.386i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.922 - 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.922 - 0.386i$
Analytic conductor: \(3.57729\)
Root analytic conductor: \(1.89137\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1/2),\ -0.922 - 0.386i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.134175 + 0.667306i\)
\(L(\frac12)\) \(\approx\) \(0.134175 + 0.667306i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.73 + 2i)T \)
good3 \( 1 + (0.866 - 1.5i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.59 - 4.5i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (2.5 - 4.33i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.866 + 1.5i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.866 + 1.5i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 8T + 29T^{2} \)
31 \( 1 + (4.33 - 7.5i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.5 + 4.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 4T + 41T^{2} \)
43 \( 1 - 6.92T + 43T^{2} \)
47 \( 1 + (-4.33 - 7.5i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.866 - 1.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.5 - 9.52i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.06 + 10.5i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 13.8T + 71T^{2} \)
73 \( 1 + (7.5 - 12.9i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.866 + 1.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.92T + 83T^{2} \)
89 \( 1 + (3.5 + 6.06i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 12T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.94554916934050521426462588954, −10.63843689479770106774165790960, −9.956747487893426506008993433473, −9.092905347740175566577899065593, −7.62944754832085948498509454762, −6.88255093177831673342426379696, −5.75083472431591313032915091470, −4.63224995481837020911237578452, −3.86031676145140149729509640890, −2.24539711565485849727439402070, 0.42684712419728731085954410879, 2.21075366775886574506629923410, 3.57133345901698184193825708254, 5.40044168577497510452821801160, 5.85868034271099397450013119908, 6.89775318776566433154205662380, 7.87157349854703519668036689047, 8.987902905040831598167703054190, 9.600727640457264582859605418302, 11.02069348719407765534732986395

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