Properties

Label 2-21e2-21.5-c1-0-8
Degree $2$
Conductor $441$
Sign $-0.652 + 0.757i$
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  + (−2.09 − 1.20i)2-s + (1.91 + 3.31i)4-s + (1.68 − 2.92i)5-s − 4.41i·8-s + (−7.06 + 4.07i)10-s + (0.717 − 0.414i)11-s − 3.37i·13-s + (−1.49 + 2.59i)16-s + (0.699 + 1.21i)17-s + (5.85 + 3.37i)19-s + 12.9·20-s − 2·22-s + (−1.73 − i)23-s + (−3.20 − 5.55i)25-s + (−4.07 + 7.06i)26-s + ⋯
L(s)  = 1  + (−1.47 − 0.853i)2-s + (0.957 + 1.65i)4-s + (0.755 − 1.30i)5-s − 1.56i·8-s + (−2.23 + 1.28i)10-s + (0.216 − 0.124i)11-s − 0.937i·13-s + (−0.374 + 0.649i)16-s + (0.169 + 0.293i)17-s + (1.34 + 0.775i)19-s + 2.89·20-s − 0.426·22-s + (−0.361 − 0.208i)23-s + (−0.641 − 1.11i)25-s + (−0.799 + 1.38i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.652 + 0.757i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.652 + 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.652 + 0.757i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.652 + 0.757i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.310452 - 0.677377i\)
\(L(\frac12)\) \(\approx\) \(0.310452 - 0.677377i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good2 \( 1 + (2.09 + 1.20i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-1.68 + 2.92i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.717 + 0.414i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.37iT - 13T^{2} \)
17 \( 1 + (-0.699 - 1.21i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.85 - 3.37i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.73 + i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.82iT - 29T^{2} \)
31 \( 1 + (-5.85 + 3.37i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.29 + 2.23i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.15T + 41T^{2} \)
43 \( 1 + 12.4T + 43T^{2} \)
47 \( 1 + (-3.37 + 5.85i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.12 - 3.53i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.37 - 5.85i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.06 + 4.07i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.24 + 7.34i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.82iT - 71T^{2} \)
73 \( 1 + (-1.21 + 0.699i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.82 + 8.36i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 13.5T + 83T^{2} \)
89 \( 1 + (-3.08 + 5.34i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 1.39iT - 97T^{2} \)
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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

−10.35692419147813790930944201746, −9.910860602151006675443537250158, −9.149319206369543980869892571645, −8.298778664151790785405886679626, −7.72846837696621014807054320108, −6.09346205752431447176184158707, −5.02803775177266211159023456039, −3.36957171998602809141615101950, −1.90149470286281857284552611909, −0.817212388203123681724801325526, 1.62046762129642473545799311290, 3.09818428821754022107271108220, 5.16737761572310307075555186904, 6.47956086431795499762091643912, 6.78739822403364521300543015888, 7.68349389679247809657183119873, 8.830408795722494714401658484399, 9.666570507663617516857236050699, 10.13651482611528955042968104806, 11.08037985916518807968738999942

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