Properties

Label 2-21e2-7.2-c3-0-22
Degree $2$
Conductor $441$
Sign $0.605 - 0.795i$
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  + (0.124 − 0.214i)2-s + (3.96 + 6.87i)4-s + (6.21 − 10.7i)5-s + 3.95·8-s + (−1.54 − 2.67i)10-s + (30.1 + 52.2i)11-s − 36.4·13-s + (−31.2 + 54.1i)16-s + (24.3 + 42.2i)17-s + (−25.2 + 43.7i)19-s + 98.7·20-s + 14.9·22-s + (69.3 − 120. i)23-s + (−14.8 − 25.6i)25-s + (−4.51 + 7.82i)26-s + ⋯
L(s)  = 1  + (0.0438 − 0.0759i)2-s + (0.496 + 0.859i)4-s + (0.556 − 0.963i)5-s + 0.174·8-s + (−0.0487 − 0.0844i)10-s + (0.826 + 1.43i)11-s − 0.777·13-s + (−0.488 + 0.846i)16-s + (0.347 + 0.602i)17-s + (−0.305 + 0.528i)19-s + 1.10·20-s + 0.144·22-s + (0.629 − 1.08i)23-s + (−0.118 − 0.205i)25-s + (−0.0340 + 0.0590i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.605 - 0.795i$
Analytic conductor: \(26.0198\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ 0.605 - 0.795i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.451732030\)
\(L(\frac12)\) \(\approx\) \(2.451732030\)
\(L(\frac{5}{2})\) 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 \( 1 + (-0.124 + 0.214i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + (-6.21 + 10.7i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-30.1 - 52.2i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 36.4T + 2.19e3T^{2} \)
17 \( 1 + (-24.3 - 42.2i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (25.2 - 43.7i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-69.3 + 120. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 61.1T + 2.43e4T^{2} \)
31 \( 1 + (0.584 + 1.01i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (34.7 - 60.2i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 308.T + 6.89e4T^{2} \)
43 \( 1 - 174.T + 7.95e4T^{2} \)
47 \( 1 + (194. - 337. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-157. - 272. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (422. + 731. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (169. - 293. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-485. - 841. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 98.4T + 3.57e5T^{2} \)
73 \( 1 + (-355. - 615. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-243. + 421. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 605.T + 5.71e5T^{2} \)
89 \( 1 + (109. - 188. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 782.T + 9.12e5T^{2} \)
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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

−10.85770878692689802962294209247, −9.815604370922232327408916985566, −9.061326356093936375414428095477, −8.102683311244486808278863021355, −7.16639794699603780489580078439, −6.25155772938929305640841138682, −4.85891579313912186079242334992, −4.08358416423898795554591788345, −2.51333192088507661411966338257, −1.43364985558437990658084809992, 0.819136056180161874467411004213, 2.30706017889915223595533478757, 3.32710009159538161233182162359, 5.03610045589832740904822569056, 6.00091572345973169077779499788, 6.66575427480643351161427449412, 7.53267978340416259501150453446, 9.037057354689826752070985032169, 9.728253307288798402332332034996, 10.70894055006593186652371255620

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