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Properties

Label 2-21e2-21.5-c3-0-31
Degree $2$
Conductor $441$
Sign $0.778 - 0.627i$
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  + (4.21 + 2.43i)2-s + (7.83 + 13.5i)4-s + (6.38 − 11.0i)5-s + 37.3i·8-s + (53.7 − 31.0i)10-s + (46.8 − 27.0i)11-s + 8.85i·13-s + (−28.1 + 48.7i)16-s + (−34.4 − 59.6i)17-s + (141. + 81.9i)19-s + 200.·20-s + 263.·22-s + (−81.3 − 46.9i)23-s + (−18.9 − 32.8i)25-s + (−21.5 + 37.3i)26-s + ⋯
L(s)  = 1  + (1.48 + 0.860i)2-s + (0.979 + 1.69i)4-s + (0.570 − 0.988i)5-s + 1.64i·8-s + (1.70 − 0.981i)10-s + (1.28 − 0.741i)11-s + 0.188i·13-s + (−0.439 + 0.760i)16-s + (−0.491 − 0.851i)17-s + (1.71 + 0.989i)19-s + 2.23·20-s + 2.55·22-s + (−0.737 − 0.425i)23-s + (−0.151 − 0.262i)25-s + (−0.162 + 0.281i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.778 - 0.627i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.778 - 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.778 - 0.627i$
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: \((2,\ 441,\ (\ :3/2),\ 0.778 - 0.627i)\)

Particular Values

\(L(2)\) \(\approx\) \(5.415832702\)
\(L(\frac12)\) \(\approx\) \(5.415832702\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.21 - 2.43i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + (-6.38 + 11.0i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-46.8 + 27.0i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 8.85iT - 2.19e3T^{2} \)
17 \( 1 + (34.4 + 59.6i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-141. - 81.9i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (81.3 + 46.9i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 119. iT - 2.43e4T^{2} \)
31 \( 1 + (85.6 - 49.4i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (47.0 - 81.5i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 259.T + 6.89e4T^{2} \)
43 \( 1 - 5.01T + 7.95e4T^{2} \)
47 \( 1 + (-28.6 + 49.6i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (407. - 235. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (112. + 195. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (370. + 213. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (81.9 + 141. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 79.8iT - 3.57e5T^{2} \)
73 \( 1 + (666. - 384. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (267. - 463. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 438.T + 5.71e5T^{2} \)
89 \( 1 + (-12.8 + 22.2i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 1.38e3iT - 9.12e5T^{2} \)
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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

−11.26196499788136285945612000892, −9.635105922301415448967274130509, −8.911721667887494909229169679207, −7.77224469590565545195755437136, −6.74207559511762653709398219607, −5.86002349831879799071384950620, −5.16889124927043765119359694324, −4.18645117827096822503497959748, −3.16390478184011578402232533444, −1.32239510987665189875527550431, 1.53879571524846070291179282857, 2.57446855419167866906624838429, 3.61158290306576451270610221206, 4.53808948457535597450610314706, 5.77601526922373001190571970902, 6.45850250895490700382952847406, 7.44139576485410945756951392085, 9.260852578131916788487663494470, 10.01527658666902161434481997434, 10.91200387003845823468996335436

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