Properties

Label 2-21e2-63.4-c1-0-6
Degree $2$
Conductor $441$
Sign $-0.968 - 0.250i$
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  + (−1.26 + 2.19i)2-s + (−1.70 + 0.300i)3-s + (−2.20 − 3.82i)4-s + 0.879·5-s + (1.5 − 4.12i)6-s + 6.10·8-s + (2.81 − 1.02i)9-s + (−1.11 + 1.92i)10-s + 3.87·11-s + (4.91 + 5.85i)12-s + (−2.72 + 4.72i)13-s + (−1.49 + 0.264i)15-s + (−3.31 + 5.74i)16-s + (0.826 − 1.43i)17-s + (−1.31 + 7.48i)18-s + (1.20 + 2.08i)19-s + ⋯
L(s)  = 1  + (−0.895 + 1.55i)2-s + (−0.984 + 0.173i)3-s + (−1.10 − 1.91i)4-s + 0.393·5-s + (0.612 − 1.68i)6-s + 2.15·8-s + (0.939 − 0.342i)9-s + (−0.352 + 0.609i)10-s + 1.16·11-s + (1.41 + 1.68i)12-s + (−0.756 + 1.30i)13-s + (−0.387 + 0.0682i)15-s + (−0.829 + 1.43i)16-s + (0.200 − 0.347i)17-s + (−0.310 + 1.76i)18-s + (0.276 + 0.479i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.968 - 0.250i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.968 - 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0730635 + 0.573345i\)
\(L(\frac12)\) \(\approx\) \(0.0730635 + 0.573345i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.70 - 0.300i)T \)
7 \( 1 \)
good2 \( 1 + (1.26 - 2.19i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 0.879T + 5T^{2} \)
11 \( 1 - 3.87T + 11T^{2} \)
13 \( 1 + (2.72 - 4.72i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.826 + 1.43i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.20 - 2.08i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 3.16T + 23T^{2} \)
29 \( 1 + (-3.02 - 5.23i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.27 + 3.94i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.27 - 3.94i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.592 - 1.02i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.0923 + 0.160i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.511 - 0.885i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.64 - 6.31i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.33 - 5.76i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.29 - 2.24i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.47 - 2.56i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.68T + 71T^{2} \)
73 \( 1 + (6.39 - 11.0i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.97 + 5.15i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.109 + 0.189i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-5.51 - 9.54i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.25 - 10.8i)T + (-48.5 + 84.0i)T^{2} \)
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

−11.44216917095541930114452751411, −10.20985835472778203779088554182, −9.502411266579388476361615818855, −8.990677774828053240270620311982, −7.56879030059443413393821810599, −6.82175618271114197410592741503, −6.19693244313385338104612223364, −5.24853283800690914433769717941, −4.27457840028778284698684265837, −1.33635310078923381695324664184, 0.64212283007056521566962026463, 1.93089288537058695160671753264, 3.36778734311101839185545425651, 4.65415282673417634060203467985, 5.90343120031207152084655717913, 7.19416373516975711698894193145, 8.208817212373027529675974570407, 9.400329304696706390481679477454, 9.956437572366119961624318949006, 10.73510132687580026383505222988

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