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Properties

Label 2-1323-63.58-c1-0-16
Degree 22
Conductor 13231323
Sign 0.713+0.701i0.713 + 0.701i
Analytic cond. 10.564210.5642
Root an. cond. 3.250263.25026
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  + 0.239·2-s − 1.94·4-s + (−0.590 − 1.02i)5-s − 0.942·8-s + (−0.141 − 0.244i)10-s + (−1.85 + 3.20i)11-s + (0.5 − 0.866i)13-s + 3.66·16-s + (3.47 + 6.01i)17-s + (0.971 − 1.68i)19-s + (1.14 + 1.98i)20-s + (−0.442 + 0.766i)22-s + (−2.80 − 4.85i)23-s + (1.80 − 3.12i)25-s + (0.119 − 0.207i)26-s + ⋯
L(s)  = 1  + 0.169·2-s − 0.971·4-s + (−0.264 − 0.457i)5-s − 0.333·8-s + (−0.0446 − 0.0774i)10-s + (−0.558 + 0.967i)11-s + (0.138 − 0.240i)13-s + 0.915·16-s + (0.841 + 1.45i)17-s + (0.222 − 0.385i)19-s + (0.256 + 0.444i)20-s + (−0.0944 + 0.163i)22-s + (−0.584 − 1.01i)23-s + (0.360 − 0.624i)25-s + (0.0234 − 0.0406i)26-s + ⋯

Functional equation

Λ(s)=(1323s/2 ΓC(s) L(s)=((0.713+0.701i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.713 + 0.701i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1323s/2 ΓC(s+1/2) L(s)=((0.713+0.701i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.713 + 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13231323    =    33⋅723^{3} \cdot 7^{2}
Sign: 0.713+0.701i0.713 + 0.701i
Analytic conductor: 10.564210.5642
Root analytic conductor: 3.250263.25026
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1323(226,⋅)\chi_{1323} (226, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1323, ( :1/2), 0.713+0.701i)(2,\ 1323,\ (\ :1/2),\ 0.713 + 0.701i)

Particular Values

L(1)L(1) ≈\approx 1.1553395991.155339599
L(12)L(\frac12) ≈\approx 1.1553395991.155339599
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
7 1 1
good2 1−0.239T+2T2 1 - 0.239T + 2T^{2}
5 1+(0.590+1.02i)T+(−2.5+4.33i)T2 1 + (0.590 + 1.02i)T + (-2.5 + 4.33i)T^{2}
11 1+(1.85−3.20i)T+(−5.5−9.52i)T2 1 + (1.85 - 3.20i)T + (-5.5 - 9.52i)T^{2}
13 1+(−0.5+0.866i)T+(−6.5−11.2i)T2 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2}
17 1+(−3.47−6.01i)T+(−8.5+14.7i)T2 1 + (-3.47 - 6.01i)T + (-8.5 + 14.7i)T^{2}
19 1+(−0.971+1.68i)T+(−9.5−16.4i)T2 1 + (-0.971 + 1.68i)T + (-9.5 - 16.4i)T^{2}
23 1+(2.80+4.85i)T+(−11.5+19.9i)T2 1 + (2.80 + 4.85i)T + (-11.5 + 19.9i)T^{2}
29 1+(−0.119−0.207i)T+(−14.5+25.1i)T2 1 + (-0.119 - 0.207i)T + (-14.5 + 25.1i)T^{2}
31 1+1.66T+31T2 1 + 1.66T + 31T^{2}
37 1+(−4.77+8.26i)T+(−18.5−32.0i)T2 1 + (-4.77 + 8.26i)T + (-18.5 - 32.0i)T^{2}
41 1+(−5.09+8.81i)T+(−20.5−35.5i)T2 1 + (-5.09 + 8.81i)T + (-20.5 - 35.5i)T^{2}
43 1+(1.11+1.92i)T+(−21.5+37.2i)T2 1 + (1.11 + 1.92i)T + (-21.5 + 37.2i)T^{2}
47 1−5.82T+47T2 1 - 5.82T + 47T^{2}
53 1+(5.80+10.0i)T+(−26.5+45.8i)T2 1 + (5.80 + 10.0i)T + (-26.5 + 45.8i)T^{2}
59 1−2.60T+59T2 1 - 2.60T + 59T^{2}
61 1−7.60T+61T2 1 - 7.60T + 61T^{2}
67 1−3.50T+67T2 1 - 3.50T + 67T^{2}
71 1+8.60T+71T2 1 + 8.60T + 71T^{2}
73 1+(−7.57−13.1i)T+(−36.5+63.2i)T2 1 + (-7.57 - 13.1i)T + (-36.5 + 63.2i)T^{2}
79 1−7.37T+79T2 1 - 7.37T + 79T^{2}
83 1+(−3.47−6.01i)T+(−41.5+71.8i)T2 1 + (-3.47 - 6.01i)T + (-41.5 + 71.8i)T^{2}
89 1+(1.37−2.37i)T+(−44.5−77.0i)T2 1 + (1.37 - 2.37i)T + (-44.5 - 77.0i)T^{2}
97 1+(−3.58−6.20i)T+(−48.5+84.0i)T2 1 + (-3.58 - 6.20i)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

−9.565926216408947908806645609260, −8.572142241661018183450106222551, −8.141585293510492404175272183235, −7.21893621266779548925389195545, −5.96904952571774913716353537214, −5.24883019518876309619515284810, −4.34456310924181443946902815707, −3.71779209579323638829133581267, −2.24953764418134427057046053530, −0.62733038170574024586408531600, 0.975115404953303915485986360416, 2.94143375880927461776752814589, 3.52184256725330665546435667056, 4.69179542635325791528519206505, 5.47455455923926382377595375092, 6.25397429037364525084099625280, 7.59230318074154993421913275532, 7.938834474345043991812028630911, 9.063726613605118100642580689380, 9.614006972044862323965711498244

Graph of the ZZ-function along the critical line