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Properties

Label 2-1323-63.25-c1-0-20
Degree 22
Conductor 13231323
Sign 0.975+0.220i0.975 + 0.220i
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  + 0.879·2-s − 1.22·4-s + (0.673 − 1.16i)5-s − 2.83·8-s + (0.592 − 1.02i)10-s + (0.826 + 1.43i)11-s + (1.68 + 2.91i)13-s − 0.0418·16-s + (0.233 − 0.405i)17-s + (1.61 + 2.79i)19-s + (−0.826 + 1.43i)20-s + (0.726 + 1.25i)22-s + (4.47 − 7.74i)23-s + (1.59 + 2.75i)25-s + (1.48 + 2.56i)26-s + ⋯
L(s)  = 1  + 0.621·2-s − 0.613·4-s + (0.301 − 0.521i)5-s − 1.00·8-s + (0.187 − 0.324i)10-s + (0.249 + 0.431i)11-s + (0.467 + 0.809i)13-s − 0.0104·16-s + (0.0567 − 0.0982i)17-s + (0.370 + 0.641i)19-s + (−0.184 + 0.320i)20-s + (0.154 + 0.268i)22-s + (0.932 − 1.61i)23-s + (0.318 + 0.551i)25-s + (0.290 + 0.503i)26-s + ⋯

Functional equation

Λ(s)=(1323s/2 ΓC(s) L(s)=((0.975+0.220i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.220i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1323s/2 ΓC(s+1/2) L(s)=((0.975+0.220i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 + 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13231323    =    33⋅723^{3} \cdot 7^{2}
Sign: 0.975+0.220i0.975 + 0.220i
Analytic conductor: 10.564210.5642
Root analytic conductor: 3.250263.25026
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1323(802,⋅)\chi_{1323} (802, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1323, ( :1/2), 0.975+0.220i)(2,\ 1323,\ (\ :1/2),\ 0.975 + 0.220i)

Particular Values

L(1)L(1) ≈\approx 2.0295747062.029574706
L(12)L(\frac12) ≈\approx 2.0295747062.029574706
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 1−0.879T+2T2 1 - 0.879T + 2T^{2}
5 1+(−0.673+1.16i)T+(−2.5−4.33i)T2 1 + (-0.673 + 1.16i)T + (-2.5 - 4.33i)T^{2}
11 1+(−0.826−1.43i)T+(−5.5+9.52i)T2 1 + (-0.826 - 1.43i)T + (-5.5 + 9.52i)T^{2}
13 1+(−1.68−2.91i)T+(−6.5+11.2i)T2 1 + (-1.68 - 2.91i)T + (-6.5 + 11.2i)T^{2}
17 1+(−0.233+0.405i)T+(−8.5−14.7i)T2 1 + (-0.233 + 0.405i)T + (-8.5 - 14.7i)T^{2}
19 1+(−1.61−2.79i)T+(−9.5+16.4i)T2 1 + (-1.61 - 2.79i)T + (-9.5 + 16.4i)T^{2}
23 1+(−4.47+7.74i)T+(−11.5−19.9i)T2 1 + (-4.47 + 7.74i)T + (-11.5 - 19.9i)T^{2}
29 1+(−3.13+5.42i)T+(−14.5−25.1i)T2 1 + (-3.13 + 5.42i)T + (-14.5 - 25.1i)T^{2}
31 1−9.23T+31T2 1 - 9.23T + 31T^{2}
37 1+(4.61+7.99i)T+(−18.5+32.0i)T2 1 + (4.61 + 7.99i)T + (-18.5 + 32.0i)T^{2}
41 1+(−1.70−2.95i)T+(−20.5+35.5i)T2 1 + (-1.70 - 2.95i)T + (-20.5 + 35.5i)T^{2}
43 1+(−2.20+3.82i)T+(−21.5−37.2i)T2 1 + (-2.20 + 3.82i)T + (-21.5 - 37.2i)T^{2}
47 1+9.35T+47T2 1 + 9.35T + 47T^{2}
53 1+(0.286−0.497i)T+(−26.5−45.8i)T2 1 + (0.286 - 0.497i)T + (-26.5 - 45.8i)T^{2}
59 1−10.3T+59T2 1 - 10.3T + 59T^{2}
61 1−7.63T+61T2 1 - 7.63T + 61T^{2}
67 1−0.596T+67T2 1 - 0.596T + 67T^{2}
71 1−0.554T+71T2 1 - 0.554T + 71T^{2}
73 1+(1.02−1.77i)T+(−36.5−63.2i)T2 1 + (1.02 - 1.77i)T + (-36.5 - 63.2i)T^{2}
79 1+2.40T+79T2 1 + 2.40T + 79T^{2}
83 1+(7.52−13.0i)T+(−41.5−71.8i)T2 1 + (7.52 - 13.0i)T + (-41.5 - 71.8i)T^{2}
89 1+(−4.54−7.86i)T+(−44.5+77.0i)T2 1 + (-4.54 - 7.86i)T + (-44.5 + 77.0i)T^{2}
97 1+(−0.949+1.64i)T+(−48.5−84.0i)T2 1 + (-0.949 + 1.64i)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

−9.552542562088576796130744768628, −8.781953651579833933758631414120, −8.248369554365298655138041841391, −6.90801745634069651221251159207, −6.18223084513904871590622588065, −5.20399056228487352996145724472, −4.53272063276787728897164247088, −3.75782377640479351629457596012, −2.49642580341731286473168079976, −0.974039639536031322478803260247, 1.03744470626005648041139332905, 2.95406513760523355890324987096, 3.37981372863964212290991161340, 4.67000187049022545446688180544, 5.37160932839038192665285468334, 6.22694358266150115256101023063, 6.99447707426703190656588965029, 8.213179536753930382631998893451, 8.782734489746828852664306773207, 9.735736950274807242377486425071

Graph of the ZZ-function along the critical line