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Properties

Label 2-21e2-63.16-c1-0-10
Degree 22
Conductor 441441
Sign 0.386+0.922i0.386 + 0.922i
Analytic cond. 3.521403.52140
Root an. cond. 1.876541.87654
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.26 − 2.19i)2-s + (1.70 + 0.300i)3-s + (−2.20 + 3.82i)4-s − 0.879·5-s + (−1.5 − 4.12i)6-s + 6.10·8-s + (2.81 + 1.02i)9-s + (1.11 + 1.92i)10-s + 3.87·11-s + (−4.91 + 5.85i)12-s + (2.72 + 4.72i)13-s + (−1.49 − 0.264i)15-s + (−3.31 − 5.74i)16-s + (−0.826 − 1.43i)17-s + (−1.31 − 7.48i)18-s + (−1.20 + 2.08i)19-s + ⋯
L(s)  = 1  + (−0.895 − 1.55i)2-s + (0.984 + 0.173i)3-s + (−1.10 + 1.91i)4-s − 0.393·5-s + (−0.612 − 1.68i)6-s + 2.15·8-s + (0.939 + 0.342i)9-s + (0.352 + 0.609i)10-s + 1.16·11-s + (−1.41 + 1.68i)12-s + (0.756 + 1.30i)13-s + (−0.387 − 0.0682i)15-s + (−0.829 − 1.43i)16-s + (−0.200 − 0.347i)17-s + (−0.310 − 1.76i)18-s + (−0.276 + 0.479i)19-s + ⋯

Functional equation

Λ(s)=(441s/2 ΓC(s) L(s)=((0.386+0.922i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(441s/2 ΓC(s+1/2) L(s)=((0.386+0.922i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 441441    =    32⋅723^{2} \cdot 7^{2}
Sign: 0.386+0.922i0.386 + 0.922i
Analytic conductor: 3.521403.52140
Root analytic conductor: 1.876541.87654
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ441(79,⋅)\chi_{441} (79, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 441, ( :1/2), 0.386+0.922i)(2,\ 441,\ (\ :1/2),\ 0.386 + 0.922i)

Particular Values

L(1)L(1) ≈\approx 0.989132−0.657953i0.989132 - 0.657953i
L(12)L(\frac12) ≈\approx 0.989132−0.657953i0.989132 - 0.657953i
L(32)L(\frac{3}{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+(−1.70−0.300i)T 1 + (-1.70 - 0.300i)T
7 1 1
good2 1+(1.26+2.19i)T+(−1+1.73i)T2 1 + (1.26 + 2.19i)T + (-1 + 1.73i)T^{2}
5 1+0.879T+5T2 1 + 0.879T + 5T^{2}
11 1−3.87T+11T2 1 - 3.87T + 11T^{2}
13 1+(−2.72−4.72i)T+(−6.5+11.2i)T2 1 + (-2.72 - 4.72i)T + (-6.5 + 11.2i)T^{2}
17 1+(0.826+1.43i)T+(−8.5+14.7i)T2 1 + (0.826 + 1.43i)T + (-8.5 + 14.7i)T^{2}
19 1+(1.20−2.08i)T+(−9.5−16.4i)T2 1 + (1.20 - 2.08i)T + (-9.5 - 16.4i)T^{2}
23 1−3.16T+23T2 1 - 3.16T + 23T^{2}
29 1+(−3.02+5.23i)T+(−14.5−25.1i)T2 1 + (-3.02 + 5.23i)T + (-14.5 - 25.1i)T^{2}
31 1+(−2.27+3.94i)T+(−15.5−26.8i)T2 1 + (-2.27 + 3.94i)T + (-15.5 - 26.8i)T^{2}
37 1+(−2.27+3.94i)T+(−18.5−32.0i)T2 1 + (-2.27 + 3.94i)T + (-18.5 - 32.0i)T^{2}
41 1+(−0.592−1.02i)T+(−20.5+35.5i)T2 1 + (-0.592 - 1.02i)T + (-20.5 + 35.5i)T^{2}
43 1+(0.0923−0.160i)T+(−21.5−37.2i)T2 1 + (0.0923 - 0.160i)T + (-21.5 - 37.2i)T^{2}
47 1+(−0.511−0.885i)T+(−23.5+40.7i)T2 1 + (-0.511 - 0.885i)T + (-23.5 + 40.7i)T^{2}
53 1+(3.64+6.31i)T+(−26.5+45.8i)T2 1 + (3.64 + 6.31i)T + (-26.5 + 45.8i)T^{2}
59 1+(3.33−5.76i)T+(−29.5−51.0i)T2 1 + (3.33 - 5.76i)T + (-29.5 - 51.0i)T^{2}
61 1+(−1.29−2.24i)T+(−30.5+52.8i)T2 1 + (-1.29 - 2.24i)T + (-30.5 + 52.8i)T^{2}
67 1+(−1.47+2.56i)T+(−33.5−58.0i)T2 1 + (-1.47 + 2.56i)T + (-33.5 - 58.0i)T^{2}
71 1+3.68T+71T2 1 + 3.68T + 71T^{2}
73 1+(−6.39−11.0i)T+(−36.5+63.2i)T2 1 + (-6.39 - 11.0i)T + (-36.5 + 63.2i)T^{2}
79 1+(−2.97−5.15i)T+(−39.5+68.4i)T2 1 + (-2.97 - 5.15i)T + (-39.5 + 68.4i)T^{2}
83 1+(−0.109+0.189i)T+(−41.5−71.8i)T2 1 + (-0.109 + 0.189i)T + (-41.5 - 71.8i)T^{2}
89 1+(5.51−9.54i)T+(−44.5−77.0i)T2 1 + (5.51 - 9.54i)T + (-44.5 - 77.0i)T^{2}
97 1+(6.25−10.8i)T+(−48.5−84.0i)T2 1 + (6.25 - 10.8i)T + (-48.5 - 84.0i)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

−11.02254309381365829901519717328, −9.820870921977605294554954910621, −9.330609183293253550038182261347, −8.607515851812954866572052428020, −7.83498905910377932038268695488, −6.60632671970344356580981990757, −4.19676205139258560891287878343, −3.86459180547307648357347082800, −2.50513848106726247676398584109, −1.38781723698542096110292637509, 1.16988591322485282112455651317, 3.36873537750804530837581543237, 4.70079234571926891255859011501, 6.11140711525917726190799891190, 6.86244500321361413650105875695, 7.75330638895902582505778299944, 8.492564635350077131706833314733, 9.004378846444553257152230690407, 9.942660681219486117946263306610, 10.88919837462206476515977179858

Graph of the ZZ-function along the critical line