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Properties

Label 2-21e2-49.37-c1-0-12
Degree 22
Conductor 441441
Sign 0.775+0.631i0.775 + 0.631i
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  + (−1.56 + 1.06i)2-s + (0.582 − 1.48i)4-s + (−1.18 − 0.366i)5-s + (2.57 + 0.588i)7-s + (−0.171 − 0.749i)8-s + (2.25 − 0.694i)10-s + (−0.0362 − 0.484i)11-s + (−5.34 − 2.57i)13-s + (−4.66 + 1.83i)14-s + (3.40 + 3.16i)16-s + (−6.06 − 0.913i)17-s + (2.70 − 4.68i)19-s + (−1.23 + 1.54i)20-s + (0.573 + 0.719i)22-s + (8.59 − 1.29i)23-s + ⋯
L(s)  = 1  + (−1.10 + 0.755i)2-s + (0.291 − 0.742i)4-s + (−0.530 − 0.163i)5-s + (0.974 + 0.222i)7-s + (−0.0604 − 0.264i)8-s + (0.711 − 0.219i)10-s + (−0.0109 − 0.145i)11-s + (−1.48 − 0.713i)13-s + (−1.24 + 0.489i)14-s + (0.851 + 0.790i)16-s + (−1.47 − 0.221i)17-s + (0.620 − 1.07i)19-s + (−0.276 + 0.346i)20-s + (0.122 + 0.153i)22-s + (1.79 − 0.270i)23-s + ⋯

Functional equation

Λ(s)=(441s/2 ΓC(s) L(s)=((0.775+0.631i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 + 0.631i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(441s/2 ΓC(s+1/2) L(s)=((0.775+0.631i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.775 + 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 441441    =    32⋅723^{2} \cdot 7^{2}
Sign: 0.775+0.631i0.775 + 0.631i
Analytic conductor: 3.521403.52140
Root analytic conductor: 1.876541.87654
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ441(37,⋅)\chi_{441} (37, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 441, ( :1/2), 0.775+0.631i)(2,\ 441,\ (\ :1/2),\ 0.775 + 0.631i)

Particular Values

L(1)L(1) ≈\approx 0.508197−0.180879i0.508197 - 0.180879i
L(12)L(\frac12) ≈\approx 0.508197−0.180879i0.508197 - 0.180879i
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+(−2.57−0.588i)T 1 + (-2.57 - 0.588i)T
good2 1+(1.56−1.06i)T+(0.730−1.86i)T2 1 + (1.56 - 1.06i)T + (0.730 - 1.86i)T^{2}
5 1+(1.18+0.366i)T+(4.13+2.81i)T2 1 + (1.18 + 0.366i)T + (4.13 + 2.81i)T^{2}
11 1+(0.0362+0.484i)T+(−10.8+1.63i)T2 1 + (0.0362 + 0.484i)T + (-10.8 + 1.63i)T^{2}
13 1+(5.34+2.57i)T+(8.10+10.1i)T2 1 + (5.34 + 2.57i)T + (8.10 + 10.1i)T^{2}
17 1+(6.06+0.913i)T+(16.2+5.01i)T2 1 + (6.06 + 0.913i)T + (16.2 + 5.01i)T^{2}
19 1+(−2.70+4.68i)T+(−9.5−16.4i)T2 1 + (-2.70 + 4.68i)T + (-9.5 - 16.4i)T^{2}
23 1+(−8.59+1.29i)T+(21.9−6.77i)T2 1 + (-8.59 + 1.29i)T + (21.9 - 6.77i)T^{2}
29 1+(−2.95+3.70i)T+(−6.45−28.2i)T2 1 + (-2.95 + 3.70i)T + (-6.45 - 28.2i)T^{2}
31 1+(1.02+1.76i)T+(−15.5+26.8i)T2 1 + (1.02 + 1.76i)T + (-15.5 + 26.8i)T^{2}
37 1+(0.441+1.12i)T+(−27.1+25.1i)T2 1 + (0.441 + 1.12i)T + (-27.1 + 25.1i)T^{2}
41 1+(−1.59−6.97i)T+(−36.9+17.7i)T2 1 + (-1.59 - 6.97i)T + (-36.9 + 17.7i)T^{2}
43 1+(−1.29+5.65i)T+(−38.7−18.6i)T2 1 + (-1.29 + 5.65i)T + (-38.7 - 18.6i)T^{2}
47 1+(−5.58+3.80i)T+(17.1−43.7i)T2 1 + (-5.58 + 3.80i)T + (17.1 - 43.7i)T^{2}
53 1+(−4.63+11.8i)T+(−38.8−36.0i)T2 1 + (-4.63 + 11.8i)T + (-38.8 - 36.0i)T^{2}
59 1+(0.334−0.103i)T+(48.7−33.2i)T2 1 + (0.334 - 0.103i)T + (48.7 - 33.2i)T^{2}
61 1+(0.0327+0.0833i)T+(−44.7+41.4i)T2 1 + (0.0327 + 0.0833i)T + (-44.7 + 41.4i)T^{2}
67 1+(−0.180−0.313i)T+(−33.5+58.0i)T2 1 + (-0.180 - 0.313i)T + (-33.5 + 58.0i)T^{2}
71 1+(4.46+5.60i)T+(−15.7+69.2i)T2 1 + (4.46 + 5.60i)T + (-15.7 + 69.2i)T^{2}
73 1+(−0.740−0.504i)T+(26.6+67.9i)T2 1 + (-0.740 - 0.504i)T + (26.6 + 67.9i)T^{2}
79 1+(6.89−11.9i)T+(−39.5−68.4i)T2 1 + (6.89 - 11.9i)T + (-39.5 - 68.4i)T^{2}
83 1+(4.54−2.18i)T+(51.7−64.8i)T2 1 + (4.54 - 2.18i)T + (51.7 - 64.8i)T^{2}
89 1+(−0.904+12.0i)T+(−88.0−13.2i)T2 1 + (-0.904 + 12.0i)T + (-88.0 - 13.2i)T^{2}
97 1+4.41T+97T2 1 + 4.41T + 97T^{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

−10.93463460383769921859270706356, −9.849152814544649821619407217425, −8.965518008065482516289230654505, −8.312565819629730050940287715703, −7.43550059753246781816658843197, −6.85450193293565967801943420330, −5.33416000179687644447770558809, −4.42400271880437435975619695095, −2.58500292114210792836752950908, −0.51341054840364928408003438351, 1.47311442655028465939390290369, 2.68348595120719367972860736258, 4.30424225625482042390513328143, 5.30762556817920219013045609196, 7.11520549966611653395064652359, 7.65921339528569372954036766310, 8.758867558454854532283863306787, 9.365920866040855493588653637900, 10.45101379651254681900346524035, 11.07439982787982432744669360381

Graph of the ZZ-function along the critical line