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Properties

Label 2-21e2-7.4-c3-0-18
Degree $2$
Conductor $441$
Sign $-0.991 + 0.126i$
Analytic cond. $26.0198$
Root an. cond. $5.10096$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2 + 3.46i)2-s + (−3.99 + 6.92i)4-s + (9 + 15.5i)5-s + (−36 + 62.3i)10-s + (−25 + 43.3i)11-s − 36·13-s + (31.9 + 55.4i)16-s + (63 − 109. i)17-s + (36 + 62.3i)19-s − 144·20-s − 200·22-s + (7 + 12.1i)23-s + (−99.5 + 172. i)25-s + (−72 − 124. i)26-s − 158·29-s + ⋯
L(s)  = 1  + (0.707 + 1.22i)2-s + (−0.499 + 0.866i)4-s + (0.804 + 1.39i)5-s + (−1.13 + 1.97i)10-s + (−0.685 + 1.18i)11-s − 0.768·13-s + (0.499 + 0.866i)16-s + (0.898 − 1.55i)17-s + (0.434 + 0.752i)19-s − 1.60·20-s − 1.93·22-s + (0.0634 + 0.109i)23-s + (−0.796 + 1.37i)25-s + (−0.543 − 0.940i)26-s − 1.01·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.991 + 0.126i$
Analytic conductor: \(26.0198\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ -0.991 + 0.126i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.987403958\)
\(L(\frac12)\) \(\approx\) \(2.987403958\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2 - 3.46i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (-9 - 15.5i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (25 - 43.3i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 36T + 2.19e3T^{2} \)
17 \( 1 + (-63 + 109. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-36 - 62.3i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-7 - 12.1i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 158T + 2.43e4T^{2} \)
31 \( 1 + (-18 + 31.1i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-81 - 140. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 270T + 6.89e4T^{2} \)
43 \( 1 + 324T + 7.95e4T^{2} \)
47 \( 1 + (36 + 62.3i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (11 - 19.0i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-234 + 405. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (396 + 685. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (116 - 200. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 734T + 3.57e5T^{2} \)
73 \( 1 + (90 - 155. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (118 + 204. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 36T + 5.71e5T^{2} \)
89 \( 1 + (-117 - 202. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 468T + 9.12e5T^{2} \)
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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

−11.11750125645526920489569439214, −9.973782429067275128168820471918, −9.707839772485549190304386758801, −7.73862628648548741588356976419, −7.35113664707516585753102203571, −6.55473114019585918765795369822, −5.54182344074111998080655328658, −4.84811686510826080903172194078, −3.28761552118328841932336690979, −2.10985879352052494052591334413, 0.75128843312636456064363221741, 1.82981204546499058838735934758, 3.03973785342368663940085066393, 4.25789809106925798375841256425, 5.28670544784905077182741335878, 5.81647534471014863245159835337, 7.67563324164051724819918980550, 8.645623040322338854191749242535, 9.590909811573335505431432400739, 10.37959236046794352065600998233

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