↑

Properties

Label 2-21e2-49.32-c1-0-17
Degree $2$
Conductor $441$
Sign $0.713 + 0.700i$
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  + (0.206 + 2.76i)2-s + (−5.60 + 0.844i)4-s + (0.116 − 0.108i)5-s + (−1.85 − 1.88i)7-s + (−2.26 − 9.90i)8-s + (0.322 + 0.299i)10-s + (−1.22 + 0.832i)11-s + (−4.48 − 2.16i)13-s + (4.82 − 5.51i)14-s + (16.0 − 4.95i)16-s + (0.262 + 0.668i)17-s + (−2.79 − 4.83i)19-s + (−0.561 + 0.704i)20-s + (−2.55 − 3.20i)22-s + (−1.20 + 3.07i)23-s + ⋯
L(s)  = 1  + (0.146 + 1.95i)2-s + (−2.80 + 0.422i)4-s + (0.0520 − 0.0483i)5-s + (−0.700 − 0.713i)7-s + (−0.799 − 3.50i)8-s + (0.101 + 0.0946i)10-s + (−0.368 + 0.251i)11-s + (−1.24 − 0.599i)13-s + (1.29 − 1.47i)14-s + (4.01 − 1.23i)16-s + (0.0636 + 0.162i)17-s + (−0.640 − 1.10i)19-s + (−0.125 + 0.157i)20-s + (−0.544 − 0.682i)22-s + (−0.251 + 0.640i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.713 + 0.700i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.713 + 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.713 + 0.700i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (424, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.713 + 0.700i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0919258 - 0.0376069i\)
\(L(\frac12)\) \(\approx\) \(0.0919258 - 0.0376069i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.85 + 1.88i)T \)
good2 \( 1 + (-0.206 - 2.76i)T + (-1.97 + 0.298i)T^{2} \)
5 \( 1 + (-0.116 + 0.108i)T + (0.373 - 4.98i)T^{2} \)
11 \( 1 + (1.22 - 0.832i)T + (4.01 - 10.2i)T^{2} \)
13 \( 1 + (4.48 + 2.16i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 + (-0.262 - 0.668i)T + (-12.4 + 11.5i)T^{2} \)
19 \( 1 + (2.79 + 4.83i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.20 - 3.07i)T + (-16.8 - 15.6i)T^{2} \)
29 \( 1 + (2.33 - 2.93i)T + (-6.45 - 28.2i)T^{2} \)
31 \( 1 + (-0.368 + 0.639i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.44 + 0.519i)T + (35.3 + 10.9i)T^{2} \)
41 \( 1 + (0.880 + 3.85i)T + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (-1.46 + 6.40i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (-0.771 - 10.3i)T + (-46.4 + 7.00i)T^{2} \)
53 \( 1 + (10.7 - 1.62i)T + (50.6 - 15.6i)T^{2} \)
59 \( 1 + (9.45 + 8.77i)T + (4.40 + 58.8i)T^{2} \)
61 \( 1 + (-3.56 - 0.536i)T + (58.2 + 17.9i)T^{2} \)
67 \( 1 + (-3.13 + 5.42i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.474 + 0.594i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-0.0447 + 0.596i)T + (-72.1 - 10.8i)T^{2} \)
79 \( 1 + (-0.0318 - 0.0551i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.67 + 3.69i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (0.703 + 0.479i)T + (32.5 + 82.8i)T^{2} \)
97 \( 1 + 10.8T + 97T^{2} \)
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

−10.67297562939861446876679590466, −9.656100803861697776386089493316, −9.097441063274862322313745536369, −7.79660159257134365013252195754, −7.31535499585451450846089210995, −6.48898822022742829452501816435, −5.40143610872215401639183187608, −4.61878403703547638012585168859, −3.39129586803919411664770450995, −0.05749042338936681716466818422, 2.04772743154443820318215570728, 2.86772333160482895197599021892, 4.07258558200073642139187025229, 5.08689219221682617175377272786, 6.21697525488532785508368028383, 8.056057965438970736231709615610, 8.937898743376481359710014246632, 9.854757501903867521795828686101, 10.23809705599409903806815305246, 11.34892217123236746375403952451

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