Properties

Label 2-21e2-441.268-c1-0-27
Degree $2$
Conductor $441$
Sign $0.997 - 0.0675i$
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  + (0.821 + 0.560i)2-s + (−1.08 + 1.35i)3-s + (−0.369 − 0.941i)4-s + (−0.0503 + 0.220i)5-s + (−1.64 + 0.507i)6-s + (1.99 − 1.73i)7-s + (0.666 − 2.91i)8-s + (−0.665 − 2.92i)9-s + (−0.164 + 0.152i)10-s + (2.39 − 1.15i)11-s + (1.67 + 0.517i)12-s + (2.29 + 1.56i)13-s + (2.61 − 0.304i)14-s + (−0.244 − 0.306i)15-s + (0.698 − 0.648i)16-s + (1.07 − 2.72i)17-s + ⋯
L(s)  = 1  + (0.580 + 0.396i)2-s + (−0.623 + 0.781i)3-s + (−0.184 − 0.470i)4-s + (−0.0225 + 0.0986i)5-s + (−0.671 + 0.206i)6-s + (0.755 − 0.655i)7-s + (0.235 − 1.03i)8-s + (−0.221 − 0.975i)9-s + (−0.0521 + 0.0483i)10-s + (0.723 − 0.348i)11-s + (0.483 + 0.149i)12-s + (0.635 + 0.433i)13-s + (0.698 − 0.0813i)14-s + (−0.0630 − 0.0790i)15-s + (0.174 − 0.162i)16-s + (0.259 − 0.661i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0675i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.997 - 0.0675i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (268, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.997 - 0.0675i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61368 + 0.0545483i\)
\(L(\frac12)\) \(\approx\) \(1.61368 + 0.0545483i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.08 - 1.35i)T \)
7 \( 1 + (-1.99 + 1.73i)T \)
good2 \( 1 + (-0.821 - 0.560i)T + (0.730 + 1.86i)T^{2} \)
5 \( 1 + (0.0503 - 0.220i)T + (-4.50 - 2.16i)T^{2} \)
11 \( 1 + (-2.39 + 1.15i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (-2.29 - 1.56i)T + (4.74 + 12.1i)T^{2} \)
17 \( 1 + (-1.07 + 2.72i)T + (-12.4 - 11.5i)T^{2} \)
19 \( 1 + (3.31 - 5.74i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.25 + 4.08i)T + (-5.11 - 22.4i)T^{2} \)
29 \( 1 + (0.700 - 0.105i)T + (27.7 - 8.54i)T^{2} \)
31 \( 1 + (-1.46 + 2.53i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (8.10 - 1.22i)T + (35.3 - 10.9i)T^{2} \)
41 \( 1 + (-11.2 + 3.47i)T + (33.8 - 23.0i)T^{2} \)
43 \( 1 + (-1.58 - 0.490i)T + (35.5 + 24.2i)T^{2} \)
47 \( 1 + (6.76 + 4.61i)T + (17.1 + 43.7i)T^{2} \)
53 \( 1 + (-1.80 - 0.271i)T + (50.6 + 15.6i)T^{2} \)
59 \( 1 + (5.56 + 1.71i)T + (48.7 + 33.2i)T^{2} \)
61 \( 1 + (1.77 - 4.51i)T + (-44.7 - 41.4i)T^{2} \)
67 \( 1 + (-0.681 + 1.18i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.06 - 1.32i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (-0.771 - 10.2i)T + (-72.1 + 10.8i)T^{2} \)
79 \( 1 + (-0.398 - 0.690i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.14 - 4.86i)T + (30.3 - 77.2i)T^{2} \)
89 \( 1 + (-1.36 + 0.929i)T + (32.5 - 82.8i)T^{2} \)
97 \( 1 + (-4.23 + 7.33i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.93539649970114243519938115790, −10.47560895365010007893346102460, −9.437121592378580078438494452102, −8.547755683523524379900302922593, −7.03579598680049040331764467054, −6.26104707430925677373967771421, −5.31879979130409897626584206492, −4.39355214899048217635026515509, −3.69689294033379597933768634859, −1.11283459568706269742338626779, 1.58659214188030921963302229811, 2.89617912357696462425872983069, 4.42879062650194505736566323879, 5.22493318973657481170972985596, 6.29520704421336494397507960519, 7.42367357250762198507412352767, 8.367272450868787291386423154065, 9.030341749431987596206717711313, 10.85506738399241869947191508945, 11.20888136520257611338113460539

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