↑

Properties

Label 2-21e2-49.37-c1-0-6
Degree $2$
Conductor $441$
Sign $0.943 - 0.330i$
Analytic cond. $3.52140$
Root an. cond. $1.87654$
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  + (1.08 − 0.736i)2-s + (−0.105 + 0.269i)4-s + (−1.01 − 0.312i)5-s + (0.746 + 2.53i)7-s + (0.666 + 2.91i)8-s + (−1.32 + 0.409i)10-s + (−0.175 − 2.33i)11-s + (6.02 + 2.90i)13-s + (2.67 + 2.19i)14-s + (2.44 + 2.27i)16-s + (4.80 + 0.724i)17-s + (−0.456 + 0.789i)19-s + (0.191 − 0.240i)20-s + (−1.91 − 2.39i)22-s + (−1.40 + 0.212i)23-s + ⋯
L(s)  = 1  + (0.764 − 0.520i)2-s + (−0.0528 + 0.134i)4-s + (−0.453 − 0.139i)5-s + (0.282 + 0.959i)7-s + (0.235 + 1.03i)8-s + (−0.419 + 0.129i)10-s + (−0.0528 − 0.705i)11-s + (1.67 + 0.804i)13-s + (0.715 + 0.586i)14-s + (0.611 + 0.567i)16-s + (1.16 + 0.175i)17-s + (−0.104 + 0.181i)19-s + (0.0428 − 0.0537i)20-s + (−0.407 − 0.511i)22-s + (−0.293 + 0.0442i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 - 0.330i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 - 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.943 - 0.330i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.943 - 0.330i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.93197 + 0.328742i\)
\(L(\frac12)\) \(\approx\) \(1.93197 + 0.328742i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-0.746 - 2.53i)T \)
good2 \( 1 + (-1.08 + 0.736i)T + (0.730 - 1.86i)T^{2} \)
5 \( 1 + (1.01 + 0.312i)T + (4.13 + 2.81i)T^{2} \)
11 \( 1 + (0.175 + 2.33i)T + (-10.8 + 1.63i)T^{2} \)
13 \( 1 + (-6.02 - 2.90i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 + (-4.80 - 0.724i)T + (16.2 + 5.01i)T^{2} \)
19 \( 1 + (0.456 - 0.789i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.40 - 0.212i)T + (21.9 - 6.77i)T^{2} \)
29 \( 1 + (4.43 - 5.55i)T + (-6.45 - 28.2i)T^{2} \)
31 \( 1 + (2.34 + 4.06i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.70 + 4.35i)T + (-27.1 + 25.1i)T^{2} \)
41 \( 1 + (-1.80 - 7.90i)T + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (-1.66 + 7.28i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (-7.17 + 4.89i)T + (17.1 - 43.7i)T^{2} \)
53 \( 1 + (-2.08 + 5.32i)T + (-38.8 - 36.0i)T^{2} \)
59 \( 1 + (5.76 - 1.77i)T + (48.7 - 33.2i)T^{2} \)
61 \( 1 + (3.59 + 9.16i)T + (-44.7 + 41.4i)T^{2} \)
67 \( 1 + (-5.85 - 10.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.12 + 8.93i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-2.98 - 2.03i)T + (26.6 + 67.9i)T^{2} \)
79 \( 1 + (-3.67 + 6.37i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-11.8 + 5.72i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (0.156 - 2.09i)T + (-88.0 - 13.2i)T^{2} \)
97 \( 1 - 4.56T + 97T^{2} \)
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

−11.42908329383661367284655605550, −10.71582266529083659358921296613, −9.154705809970789109221757648327, −8.475718208148702642961442843233, −7.73432874456392523414286792995, −6.07170919882189962060159155866, −5.39432605830255573221928066283, −4.02749637449435520857356373017, −3.37494769273430980907416611401, −1.86726884785384950722704867446, 1.15056007164111328565089830122, 3.52375480294969109974841090195, 4.18215295065801406434907856035, 5.37668340756331508904305693884, 6.23060434729834374866446483949, 7.37641941882142434266497378075, 7.923823298522141024599186386584, 9.357906821847421318770331193199, 10.35443034400020061157529781253, 10.94348101371349940181084574749

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