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Properties

Label 2-21e2-21.17-c3-0-34
Degree $2$
Conductor $441$
Sign $0.462 + 0.886i$
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  + (4.66 − 2.69i)2-s + (10.4 − 18.1i)4-s + (9.16 + 15.8i)5-s − 69.8i·8-s + (85.4 + 49.3i)10-s + (20.5 + 11.8i)11-s + 11.7i·13-s + (−104. − 180. i)16-s + (48.2 − 83.5i)17-s + (−18.1 + 10.5i)19-s + 384.·20-s + 127.·22-s + (132. − 76.3i)23-s + (−105. + 182. i)25-s + (31.7 + 54.9i)26-s + ⋯
L(s)  = 1  + (1.64 − 0.951i)2-s + (1.31 − 2.27i)4-s + (0.819 + 1.41i)5-s − 3.08i·8-s + (2.70 + 1.55i)10-s + (0.564 + 0.325i)11-s + 0.251i·13-s + (−1.62 − 2.81i)16-s + (0.687 − 1.19i)17-s + (−0.219 + 0.126i)19-s + 4.29·20-s + 1.23·22-s + (1.19 − 0.691i)23-s + (−0.843 + 1.46i)25-s + (0.239 + 0.414i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.462 + 0.886i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.462 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.462 + 0.886i$
Analytic conductor: \(26.0198\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ 0.462 + 0.886i)\)

Particular Values

\(L(2)\) \(\approx\) \(6.139859149\)
\(L(\frac12)\) \(\approx\) \(6.139859149\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.66 + 2.69i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + (-9.16 - 15.8i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-20.5 - 11.8i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 11.7iT - 2.19e3T^{2} \)
17 \( 1 + (-48.2 + 83.5i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (18.1 - 10.5i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-132. + 76.3i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 77.5iT - 2.43e4T^{2} \)
31 \( 1 + (-172. - 99.8i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (82.1 + 142. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 435.T + 6.89e4T^{2} \)
43 \( 1 + 106.T + 7.95e4T^{2} \)
47 \( 1 + (5.52 + 9.56i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (96.0 + 55.4i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (414. - 717. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (19.5 - 11.2i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (199. - 345. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 228. iT - 3.57e5T^{2} \)
73 \( 1 + (231. + 133. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (460. + 797. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 642.T + 5.71e5T^{2} \)
89 \( 1 + (222. + 385. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.61e3iT - 9.12e5T^{2} \)
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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

−10.63709486668563189738222530774, −10.22139410454680213563041021963, −9.247278236552373899286828192053, −7.04219701492685123515064790016, −6.62548543729858363601817882972, −5.58452659011620331366169963146, −4.62815429440387831149448615056, −3.28672764475089338545321708633, −2.68189919122405876616650446798, −1.48321752301691002313367403886, 1.56071430867131388700556604233, 3.22093265532745518626745078174, 4.35739035764672912551393360184, 5.19767115372637835734662163764, 5.88783093464355167047853638615, 6.70738926626878771725114737673, 8.051644739280804196552336837499, 8.639393301787352017521171495386, 9.854430552247976947645332694053, 11.33000340336744812681567224911

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