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Properties

Label 2-21e2-21.5-c3-0-12
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  + (1.40 + 0.810i)2-s + (−2.68 − 4.65i)4-s + (−2.35 + 4.08i)5-s − 21.6i·8-s + (−6.62 + 3.82i)10-s + (−25.9 + 14.9i)11-s + 27.1i·13-s + (−3.92 + 6.80i)16-s + (−8.92 − 15.4i)17-s + (107. + 61.9i)19-s + 25.3·20-s − 48.6·22-s + (71.0 + 41.0i)23-s + (51.3 + 88.9i)25-s + (−22.0 + 38.1i)26-s + ⋯
L(s)  = 1  + (0.496 + 0.286i)2-s + (−0.335 − 0.581i)4-s + (−0.211 + 0.365i)5-s − 0.957i·8-s + (−0.209 + 0.120i)10-s + (−0.712 + 0.411i)11-s + 0.579i·13-s + (−0.0613 + 0.106i)16-s + (−0.127 − 0.220i)17-s + (1.29 + 0.747i)19-s + 0.283·20-s − 0.471·22-s + (0.644 + 0.372i)23-s + (0.410 + 0.711i)25-s + (−0.166 + 0.287i)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: $\chi_{441} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ 0.462 - 0.886i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.843603396\)
\(L(\frac12)\) \(\approx\) \(1.843603396\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.40 - 0.810i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + (2.35 - 4.08i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (25.9 - 14.9i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 27.1iT - 2.19e3T^{2} \)
17 \( 1 + (8.92 + 15.4i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-107. - 61.9i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-71.0 - 41.0i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 88.2iT - 2.43e4T^{2} \)
31 \( 1 + (-220. + 127. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (107. - 186. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 427.T + 6.89e4T^{2} \)
43 \( 1 - 62.4T + 7.95e4T^{2} \)
47 \( 1 + (211. - 366. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (587. - 339. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-383. - 664. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-313. - 180. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-323. - 561. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 536. iT - 3.57e5T^{2} \)
73 \( 1 + (-547. + 315. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-298. + 517. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 591.T + 5.71e5T^{2} \)
89 \( 1 + (769. - 1.33e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 654. iT - 9.12e5T^{2} \)
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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

−10.85622406459297651028188596356, −9.858183261826471626166927696025, −9.311718079321543526561901137004, −7.899257700239916426281128568025, −7.06718716276126533398681907555, −6.04120082278270453558860909848, −5.11187414152050136712217341627, −4.19317066421477251932247177617, −2.89222661184888032315572854272, −1.16125365037718837362637751068, 0.60090529837264498743629800208, 2.61621519735134361885144189082, 3.48136637867089757373020413118, 4.78019420137791706467020717191, 5.36513974893515980214486591928, 6.87643088764510080629156155160, 8.017168006972866861959031549812, 8.544506399786175957356000079293, 9.614997850961116407447679524600, 10.80768973424695668286233623709

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