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Properties

Label 2-21e2-21.5-c3-0-27
Degree $2$
Conductor $441$
Sign $0.652 - 0.757i$
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  + (3.21 + 1.85i)2-s + (2.87 + 4.98i)4-s + (−2.27 + 3.93i)5-s − 8.32i·8-s + (−14.6 + 8.43i)10-s + (39.4 − 22.7i)11-s − 11.6i·13-s + (38.4 − 66.6i)16-s + (45.7 + 79.2i)17-s + (121. + 70.0i)19-s − 26.1·20-s + 168.·22-s + (38.9 + 22.5i)23-s + (52.1 + 90.3i)25-s + (21.6 − 37.4i)26-s + ⋯
L(s)  = 1  + (1.13 + 0.655i)2-s + (0.359 + 0.623i)4-s + (−0.203 + 0.352i)5-s − 0.367i·8-s + (−0.461 + 0.266i)10-s + (1.08 − 0.623i)11-s − 0.249i·13-s + (0.600 − 1.04i)16-s + (0.652 + 1.13i)17-s + (1.46 + 0.845i)19-s − 0.292·20-s + 1.63·22-s + (0.353 + 0.204i)23-s + (0.417 + 0.722i)25-s + (0.163 − 0.282i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.652 - 0.757i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.652 - 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.652 - 0.757i$
Analytic conductor: \(26.0198\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ 0.652 - 0.757i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.781847043\)
\(L(\frac12)\) \(\approx\) \(3.781847043\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.21 - 1.85i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + (2.27 - 3.93i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-39.4 + 22.7i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 11.6iT - 2.19e3T^{2} \)
17 \( 1 + (-45.7 - 79.2i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-121. - 70.0i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-38.9 - 22.5i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 47.7iT - 2.43e4T^{2} \)
31 \( 1 + (206. - 119. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-74.4 + 128. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 393.T + 6.89e4T^{2} \)
43 \( 1 - 412.T + 7.95e4T^{2} \)
47 \( 1 + (-204. + 353. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (144. - 83.5i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-68.7 - 119. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-280. - 161. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (212. + 367. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 727. iT - 3.57e5T^{2} \)
73 \( 1 + (-949. + 548. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-334. + 579. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 199.T + 5.71e5T^{2} \)
89 \( 1 + (403. - 699. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 701. iT - 9.12e5T^{2} \)
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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

−10.99148872856764779841647615589, −9.958279096388695968638819471678, −8.955927074817067721522944801516, −7.71115981065578702931931870160, −6.93642315308120318180073744124, −5.91376260147763119470556553863, −5.30417214118546393913046221161, −3.80704089786165602176118953644, −3.38655149275237709565032597777, −1.21167179443239230970810547194, 1.11571835531745166919257204376, 2.59883906019899858281720103591, 3.67440708870814354293200050135, 4.66748557925301477528681400340, 5.36711500560497311581352306266, 6.71694596199999858807918593574, 7.69941697602927160732518042443, 9.004732843355958660582009436108, 9.680094231097125255604176116226, 11.01958499426203209321338243232

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