↑

Properties

Label 2-446-223.183-c1-0-8
Degree $2$
Conductor $446$
Sign $0.380 - 0.924i$
Analytic cond. $3.56132$
Root an. cond. $1.88714$
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  + 2-s + (1.48 + 2.57i)3-s + 4-s + (0.877 − 1.52i)5-s + (1.48 + 2.57i)6-s − 2.19·7-s + 8-s + (−2.92 + 5.06i)9-s + (0.877 − 1.52i)10-s + (−1.63 + 2.82i)11-s + (1.48 + 2.57i)12-s + 3.36·13-s − 2.19·14-s + 5.22·15-s + 16-s + 1.99·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.858 + 1.48i)3-s + 0.5·4-s + (0.392 − 0.679i)5-s + (0.607 + 1.05i)6-s − 0.829·7-s + 0.353·8-s + (−0.975 + 1.68i)9-s + (0.277 − 0.480i)10-s + (−0.492 + 0.852i)11-s + (0.429 + 0.743i)12-s + 0.934·13-s − 0.586·14-s + 1.34·15-s + 0.250·16-s + 0.482·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 446 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.380 - 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 446 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.380 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(446\)    =    \(2 \cdot 223\)
Sign: $0.380 - 0.924i$
Analytic conductor: \(3.56132\)
Root analytic conductor: \(1.88714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{446} (183, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 446,\ (\ :1/2),\ 0.380 - 0.924i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.22003 + 1.48702i\)
\(L(\frac12)\) \(\approx\) \(2.22003 + 1.48702i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
223 \( 1 + (-7.27 + 13.0i)T \)
good3 \( 1 + (-1.48 - 2.57i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.877 + 1.52i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 2.19T + 7T^{2} \)
11 \( 1 + (1.63 - 2.82i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.36T + 13T^{2} \)
17 \( 1 - 1.99T + 17T^{2} \)
19 \( 1 + (0.398 + 0.689i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.46 + 2.53i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.23 + 9.06i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.75 + 8.24i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.231 - 0.401i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 1.73T + 41T^{2} \)
43 \( 1 + (4.01 + 6.95i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.74 - 8.21i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.33 + 7.51i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 5.81T + 59T^{2} \)
61 \( 1 + (-5.10 - 8.84i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.22 - 2.11i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.30 + 7.45i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.35 + 9.28i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.32 - 5.76i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.728 + 1.26i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.66 - 9.81i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.71 - 8.16i)T + (-48.5 - 84.0i)T^{2} \)
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

−11.11363382760130771889871680363, −10.09288120099957464044136647954, −9.691535008021721709338592543628, −8.754968277318093781233714588066, −7.81404738959838166627338737089, −6.28244254906737025674126678077, −5.24753836621414326728103056211, −4.35525644671525085134781907814, −3.51961359966639729842736813538, −2.35977691719923661025481435169, 1.51633594983842464006610994692, 3.00200316392190915963279668852, 3.33690328845407795608847814792, 5.53911125224953324855218939900, 6.50817383501810193284437313278, 6.91212332424449949433434008274, 8.090728535478557444926569452191, 8.819916257641190114369492674236, 10.17410090472456791667956351723, 11.07440006941942714591763190807

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