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Properties

Label 2-21e2-7.2-c3-0-24
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  + (1.32 − 2.29i)2-s + (0.5 + 0.866i)4-s + 23.8·8-s + (13.2 + 22.9i)11-s + (27.5 − 47.6i)16-s + 70·22-s + (−108. + 187. i)23-s + (62.5 + 108. i)25-s + 264.·29-s + (22.4 + 38.9i)32-s + (225 − 389. i)37-s + 180·43-s + (−13.2 + 22.9i)44-s + (287 + 497. i)46-s + 330.·50-s + ⋯
L(s)  = 1  + (0.467 − 0.810i)2-s + (0.0625 + 0.108i)4-s + 1.05·8-s + (0.362 + 0.628i)11-s + (0.429 − 0.744i)16-s + 0.678·22-s + (−0.983 + 1.70i)23-s + (0.5 + 0.866i)25-s + 1.69·29-s + (0.124 + 0.215i)32-s + (0.999 − 1.73i)37-s + 0.638·43-s + (−0.0453 + 0.0785i)44-s + (0.919 + 1.59i)46-s + 0.935·50-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: $\chi_{441} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ 0.991 + 0.126i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.877145539\)
\(L(\frac12)\) \(\approx\) \(2.877145539\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.32 + 2.29i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-13.2 - 22.9i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 2.19e3T^{2} \)
17 \( 1 + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (108. - 187. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 264.T + 2.43e4T^{2} \)
31 \( 1 + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-225 + 389. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 6.89e4T^{2} \)
43 \( 1 - 180T + 7.95e4T^{2} \)
47 \( 1 + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-248. - 430. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-370 - 640. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 978.T + 3.57e5T^{2} \)
73 \( 1 + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-692 + 1.19e3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 5.71e5T^{2} \)
89 \( 1 + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 9.12e5T^{2} \)
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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

−10.85395710417885533715223073155, −9.996799822039791777150679102738, −9.069732091076104507659871909327, −7.78869916312217894240066233442, −7.12082316789439417638092366455, −5.77983622001850314864533994394, −4.55410936206650880031490132760, −3.69044131594193917848776751902, −2.51967202906074956288802097086, −1.33382297849119120412243813919, 0.910214200376569666844401432401, 2.56711111153160769481502607587, 4.17024369324560223942935646878, 5.00202026899963173516861191815, 6.32169442387289091295253546696, 6.53973108056831443657152726811, 7.952580460436214949712366220357, 8.586759091349260183240127129927, 9.999203402250988396625273596966, 10.60132079527576410236941844352

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