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Properties

Label 2-21e2-63.13-c2-0-50
Degree 22
Conductor 441441
Sign 0.940−0.339i0.940 - 0.339i
Analytic cond. 12.016312.0163
Root an. cond. 3.466463.46646
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.902 + 1.56i)2-s + (2.28 − 1.94i)3-s + (0.371 − 0.643i)4-s + (4.98 + 2.87i)5-s + (5.09 + 1.81i)6-s + 8.56·8-s + (1.44 − 8.88i)9-s + 10.3i·10-s + (5.09 + 8.82i)11-s + (−0.401 − 2.19i)12-s + (−9.76 − 5.63i)13-s + (16.9 − 3.11i)15-s + (6.23 + 10.8i)16-s + 17.7i·17-s + (15.1 − 5.75i)18-s − 30.7i·19-s + ⋯
L(s)  = 1  + (0.451 + 0.781i)2-s + (0.761 − 0.647i)3-s + (0.0929 − 0.160i)4-s + (0.996 + 0.575i)5-s + (0.849 + 0.302i)6-s + 1.07·8-s + (0.160 − 0.987i)9-s + 1.03i·10-s + (0.463 + 0.802i)11-s + (−0.0334 − 0.182i)12-s + (−0.750 − 0.433i)13-s + (1.13 − 0.207i)15-s + (0.389 + 0.675i)16-s + 1.04i·17-s + (0.843 − 0.319i)18-s − 1.61i·19-s + ⋯

Functional equation

Λ(s)=(441s/2 ΓC(s) L(s)=((0.940−0.339i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 - 0.339i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(441s/2 ΓC(s+1) L(s)=((0.940−0.339i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.940 - 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 441441    =    32⋅723^{2} \cdot 7^{2}
Sign: 0.940−0.339i0.940 - 0.339i
Analytic conductor: 12.016312.0163
Root analytic conductor: 3.466463.46646
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ441(391,⋅)\chi_{441} (391, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 441, ( :1), 0.940−0.339i)(2,\ 441,\ (\ :1),\ 0.940 - 0.339i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 3.60389+0.631153i3.60389 + 0.631153i
L(12)L(\frac12) ≈\approx 3.60389+0.631153i3.60389 + 0.631153i
L(2)L(2) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1+(−2.28+1.94i)T 1 + (-2.28 + 1.94i)T
7 1 1
good2 1+(−0.902−1.56i)T+(−2+3.46i)T2 1 + (-0.902 - 1.56i)T + (-2 + 3.46i)T^{2}
5 1+(−4.98−2.87i)T+(12.5+21.6i)T2 1 + (-4.98 - 2.87i)T + (12.5 + 21.6i)T^{2}
11 1+(−5.09−8.82i)T+(−60.5+104.i)T2 1 + (-5.09 - 8.82i)T + (-60.5 + 104. i)T^{2}
13 1+(9.76+5.63i)T+(84.5+146.i)T2 1 + (9.76 + 5.63i)T + (84.5 + 146. i)T^{2}
17 1−17.7iT−289T2 1 - 17.7iT - 289T^{2}
19 1+30.7iT−361T2 1 + 30.7iT - 361T^{2}
23 1+(15.6−27.1i)T+(−264.5−458.i)T2 1 + (15.6 - 27.1i)T + (-264.5 - 458. i)T^{2}
29 1+(−1.48−2.56i)T+(−420.5+728.i)T2 1 + (-1.48 - 2.56i)T + (-420.5 + 728. i)T^{2}
31 1+(25.0+14.4i)T+(480.5+832.i)T2 1 + (25.0 + 14.4i)T + (480.5 + 832. i)T^{2}
37 1−22.1T+1.36e3T2 1 - 22.1T + 1.36e3T^{2}
41 1+(9.95+5.74i)T+(840.5+1.45e3i)T2 1 + (9.95 + 5.74i)T + (840.5 + 1.45e3i)T^{2}
43 1+(−19.6−34.0i)T+(−924.5+1.60e3i)T2 1 + (-19.6 - 34.0i)T + (-924.5 + 1.60e3i)T^{2}
47 1+(−18.5+10.7i)T+(1.10e3−1.91e3i)T2 1 + (-18.5 + 10.7i)T + (1.10e3 - 1.91e3i)T^{2}
53 1−45.7T+2.80e3T2 1 - 45.7T + 2.80e3T^{2}
59 1+(11.1+6.42i)T+(1.74e3+3.01e3i)T2 1 + (11.1 + 6.42i)T + (1.74e3 + 3.01e3i)T^{2}
61 1+(72.4−41.8i)T+(1.86e3−3.22e3i)T2 1 + (72.4 - 41.8i)T + (1.86e3 - 3.22e3i)T^{2}
67 1+(10.1−17.6i)T+(−2.24e3−3.88e3i)T2 1 + (10.1 - 17.6i)T + (-2.24e3 - 3.88e3i)T^{2}
71 1+92.6T+5.04e3T2 1 + 92.6T + 5.04e3T^{2}
73 1+102.iT−5.32e3T2 1 + 102. iT - 5.32e3T^{2}
79 1+(−50.2−86.9i)T+(−3.12e3+5.40e3i)T2 1 + (-50.2 - 86.9i)T + (-3.12e3 + 5.40e3i)T^{2}
83 1+(20.6−11.9i)T+(3.44e3−5.96e3i)T2 1 + (20.6 - 11.9i)T + (3.44e3 - 5.96e3i)T^{2}
89 1−55.9iT−7.92e3T2 1 - 55.9iT - 7.92e3T^{2}
97 1+(69.6−40.1i)T+(4.70e3−8.14e3i)T2 1 + (69.6 - 40.1i)T + (4.70e3 - 8.14e3i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−10.81445066233557405504981959314, −9.878753402027952652118937930064, −9.223847258314558447721006108984, −7.80738759438550230511678773928, −7.13093411518222286244029726893, −6.38372832528349539492302656506, −5.55091990639041942216534537708, −4.19650909050858453762268110266, −2.57237446596611253480646038612, −1.62433090957237934379620413627, 1.69031134677528799233598273333, 2.67014260156225410242237071249, 3.81461364961463862361841066010, 4.74076466837563007774075199459, 5.81718979087745998081801550709, 7.31888993481147057915910996687, 8.360621405822654767526259570727, 9.252077005927362702250456598758, 10.00831404195258627075516953429, 10.75079640648091928157835954048

Graph of the ZZ-function along the critical line