Properties

Label 2-21e2-63.5-c1-0-7
Degree $2$
Conductor $441$
Sign $-0.235 - 0.971i$
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  + 1.73i·2-s − 1.73i·3-s − 0.999·4-s + (1.5 + 2.59i)5-s + 2.99·6-s + 1.73i·8-s − 2.99·9-s + (−4.5 + 2.59i)10-s + (1.5 + 0.866i)11-s + 1.73i·12-s + (−1.5 − 0.866i)13-s + (4.5 − 2.59i)15-s − 5·16-s + (1.5 + 2.59i)17-s − 5.19i·18-s + (4.5 + 2.59i)19-s + ⋯
L(s)  = 1  + 1.22i·2-s − 0.999i·3-s − 0.499·4-s + (0.670 + 1.16i)5-s + 1.22·6-s + 0.612i·8-s − 0.999·9-s + (−1.42 + 0.821i)10-s + (0.452 + 0.261i)11-s + 0.499i·12-s + (−0.416 − 0.240i)13-s + (1.16 − 0.670i)15-s − 1.25·16-s + (0.363 + 0.630i)17-s − 1.22i·18-s + (1.03 + 0.596i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.235 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.235 - 0.971i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.235 - 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.975372 + 1.24017i\)
\(L(\frac12)\) \(\approx\) \(0.975372 + 1.24017i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.73iT \)
7 \( 1 \)
good2 \( 1 - 1.73iT - 2T^{2} \)
5 \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.5 - 0.866i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.5 + 0.866i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.5 - 2.59i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.5 + 2.59i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.5 - 2.59i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 + (3.5 - 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (-7.5 + 4.33i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 13.8iT - 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + 3.46iT - 71T^{2} \)
73 \( 1 + (-4.5 + 2.59i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + (7.5 + 12.9i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.5 + 2.59i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.5 - 0.866i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.40632432733225108182000205329, −10.50328884458875860361275540767, −9.367751134025422031806863187691, −8.240041641280609738669055701456, −7.37493504212361249897340938760, −6.80947594156380770543209143256, −6.06458625881309539539596987073, −5.25768247432087086051004324024, −3.17040035285208912741599036083, −1.94081127087227006539882941294, 1.08390794351730416373759210551, 2.64444307993536269557638908476, 3.78191481176377482405687732830, 4.87761163420525156942783335090, 5.63165539068504427278782897037, 7.20026455941060238563394181494, 8.841072049975956968599589901107, 9.347695112760174966469296447698, 9.861328490816099585712786417665, 10.92516608470411466306902315438

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