Properties

Label 2-21e2-49.4-c1-0-8
Degree $2$
Conductor $441$
Sign $0.364 + 0.931i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.52 − 1.04i)2-s + (0.514 + 1.31i)4-s + (1.32 − 0.410i)5-s + (−1.73 + 1.99i)7-s + (−0.242 + 1.06i)8-s + (−2.45 − 0.757i)10-s + (0.215 − 2.87i)11-s + (4.03 − 1.94i)13-s + (4.72 − 1.23i)14-s + (3.54 − 3.28i)16-s + (0.476 − 0.0718i)17-s + (1.90 + 3.30i)19-s + (1.22 + 1.53i)20-s + (−3.31 + 4.16i)22-s + (6.73 + 1.01i)23-s + ⋯
L(s)  = 1  + (−1.07 − 0.735i)2-s + (0.257 + 0.655i)4-s + (0.594 − 0.183i)5-s + (−0.656 + 0.754i)7-s + (−0.0859 + 0.376i)8-s + (−0.776 − 0.239i)10-s + (0.0649 − 0.866i)11-s + (1.12 − 0.539i)13-s + (1.26 − 0.330i)14-s + (0.885 − 0.821i)16-s + (0.115 − 0.0174i)17-s + (0.437 + 0.757i)19-s + (0.273 + 0.342i)20-s + (−0.707 + 0.887i)22-s + (1.40 + 0.211i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.364 + 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.364 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.364 + 0.931i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (298, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ 0.364 + 0.931i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.695815 - 0.474992i\)
\(L(\frac12)\) \(\approx\) \(0.695815 - 0.474992i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.73 - 1.99i)T \)
good2 \( 1 + (1.52 + 1.04i)T + (0.730 + 1.86i)T^{2} \)
5 \( 1 + (-1.32 + 0.410i)T + (4.13 - 2.81i)T^{2} \)
11 \( 1 + (-0.215 + 2.87i)T + (-10.8 - 1.63i)T^{2} \)
13 \( 1 + (-4.03 + 1.94i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 + (-0.476 + 0.0718i)T + (16.2 - 5.01i)T^{2} \)
19 \( 1 + (-1.90 - 3.30i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.73 - 1.01i)T + (21.9 + 6.77i)T^{2} \)
29 \( 1 + (1.45 + 1.81i)T + (-6.45 + 28.2i)T^{2} \)
31 \( 1 + (-3.94 + 6.82i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.50 + 8.92i)T + (-27.1 - 25.1i)T^{2} \)
41 \( 1 + (-1.56 + 6.84i)T + (-36.9 - 17.7i)T^{2} \)
43 \( 1 + (-0.546 - 2.39i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (-9.25 - 6.30i)T + (17.1 + 43.7i)T^{2} \)
53 \( 1 + (1.96 + 4.99i)T + (-38.8 + 36.0i)T^{2} \)
59 \( 1 + (3.57 + 1.10i)T + (48.7 + 33.2i)T^{2} \)
61 \( 1 + (-0.0147 + 0.0376i)T + (-44.7 - 41.4i)T^{2} \)
67 \( 1 + (0.534 - 0.926i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.21 - 5.28i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (6.78 - 4.62i)T + (26.6 - 67.9i)T^{2} \)
79 \( 1 + (-6.91 - 11.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.465 - 0.224i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (0.0586 + 0.782i)T + (-88.0 + 13.2i)T^{2} \)
97 \( 1 - 9.61T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.90912303331941113761791097094, −9.898543969461108377862942029344, −9.234829177527490896028373904907, −8.633283420618070922424868070098, −7.64501496480909435322036197855, −5.94327646676321701183817557471, −5.63071110975940362927116049997, −3.53664357355001658535777622051, −2.43956809169151425121809480243, −0.967917335077878519646490341673, 1.20492727604732733487362373073, 3.19345675901726979667407833394, 4.57027411136076427604850977234, 6.17335516970631950254977896737, 6.79356689576163702282258666406, 7.50565247837047245445902413494, 8.716929211879700654120856621069, 9.376306137996913114730353109660, 10.13657707096604265330351989484, 10.84656343725082568649842704876

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