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Properties

Label 2-21e2-21.20-c5-0-29
Degree $2$
Conductor $441$
Sign $-0.970 - 0.239i$
Analytic cond. $70.7292$
Root an. cond. $8.41006$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 10.9i·2-s − 88.0·4-s − 82.6·5-s + 614. i·8-s + 905. i·10-s + 113. i·11-s − 329. i·13-s + 3.91e3·16-s + 443.·17-s + 1.67e3i·19-s + 7.27e3·20-s + 1.24e3·22-s + 224. i·23-s + 3.69e3·25-s − 3.60e3·26-s + ⋯
L(s)  = 1  − 1.93i·2-s − 2.75·4-s − 1.47·5-s + 3.39i·8-s + 2.86i·10-s + 0.283i·11-s − 0.540i·13-s + 3.82·16-s + 0.371·17-s + 1.06i·19-s + 4.06·20-s + 0.549·22-s + 0.0883i·23-s + 1.18·25-s − 1.04·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 - 0.239i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.970 - 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.970 - 0.239i$
Analytic conductor: \(70.7292\)
Root analytic conductor: \(8.41006\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (440, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :5/2),\ -0.970 - 0.239i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.6447695321\)
\(L(\frac12)\) \(\approx\) \(0.6447695321\)
\(L(\frac{7}{2})\) 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 \( 1 + 10.9iT - 32T^{2} \)
5 \( 1 + 82.6T + 3.12e3T^{2} \)
11 \( 1 - 113. iT - 1.61e5T^{2} \)
13 \( 1 + 329. iT - 3.71e5T^{2} \)
17 \( 1 - 443.T + 1.41e6T^{2} \)
19 \( 1 - 1.67e3iT - 2.47e6T^{2} \)
23 \( 1 - 224. iT - 6.43e6T^{2} \)
29 \( 1 + 2.98e3iT - 2.05e7T^{2} \)
31 \( 1 - 3.37e3iT - 2.86e7T^{2} \)
37 \( 1 + 1.17e4T + 6.93e7T^{2} \)
41 \( 1 - 1.45e4T + 1.15e8T^{2} \)
43 \( 1 + 1.05e4T + 1.47e8T^{2} \)
47 \( 1 - 2.20e4T + 2.29e8T^{2} \)
53 \( 1 - 4.55e3iT - 4.18e8T^{2} \)
59 \( 1 + 1.33e4T + 7.14e8T^{2} \)
61 \( 1 - 5.07e4iT - 8.44e8T^{2} \)
67 \( 1 + 5.13e4T + 1.35e9T^{2} \)
71 \( 1 - 1.20e3iT - 1.80e9T^{2} \)
73 \( 1 - 8.83e3iT - 2.07e9T^{2} \)
79 \( 1 - 4.35e4T + 3.07e9T^{2} \)
83 \( 1 + 5.81e4T + 3.93e9T^{2} \)
89 \( 1 - 1.08e5T + 5.58e9T^{2} \)
97 \( 1 - 5.44e3iT - 8.58e9T^{2} \)
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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

−10.26313799323145863412004664824, −9.130222581772584475697859307323, −8.262446258869260327752506270437, −7.52550318043427036523629661125, −5.53334892272200080848409796984, −4.37379044537801079815815245806, −3.71400187019006818849050475955, −2.82729115073904653920230655060, −1.41590143749142754342862896946, −0.29569959006997100935429218750, 0.63813010922284791997541466371, 3.46869787731464779337590936155, 4.35124218696743552429307381855, 5.18206709369200022919590684084, 6.38530202419774157585085103692, 7.22211009617648996717705373153, 7.80331997143808489156758590158, 8.678977974461227702984812879108, 9.310760620840040204978046529227, 10.69876838110371514722252020794

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