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Properties

Label 2-1323-63.58-c1-0-25
Degree 22
Conductor 13231323
Sign −0.823+0.566i-0.823 + 0.566i
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  − 1.34·2-s − 0.184·4-s + (−1.26 − 2.19i)5-s + 2.94·8-s + (1.70 + 2.95i)10-s + (0.233 − 0.405i)11-s + (2.91 − 5.04i)13-s − 3.59·16-s + (−1.93 − 3.35i)17-s + (−1.09 + 1.89i)19-s + (0.233 + 0.405i)20-s + (−0.315 + 0.545i)22-s + (−0.0530 − 0.0918i)23-s + (−0.705 + 1.22i)25-s + (−3.92 + 6.79i)26-s + ⋯
L(s)  = 1  − 0.952·2-s − 0.0923·4-s + (−0.566 − 0.980i)5-s + 1.04·8-s + (0.539 + 0.934i)10-s + (0.0705 − 0.122i)11-s + (0.807 − 1.39i)13-s − 0.899·16-s + (−0.470 − 0.814i)17-s + (−0.250 + 0.434i)19-s + (0.0523 + 0.0906i)20-s + (−0.0672 + 0.116i)22-s + (−0.0110 − 0.0191i)23-s + (−0.141 + 0.244i)25-s + (−0.769 + 1.33i)26-s + ⋯

Functional equation

Λ(s)=(1323s/2 ΓC(s) L(s)=((−0.823+0.566i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.823 + 0.566i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1323s/2 ΓC(s+1/2) L(s)=((−0.823+0.566i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.823 + 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13231323    =    33⋅723^{3} \cdot 7^{2}
Sign: −0.823+0.566i-0.823 + 0.566i
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), −0.823+0.566i)(2,\ 1323,\ (\ :1/2),\ -0.823 + 0.566i)

Particular Values

L(1)L(1) ≈\approx 0.55774253140.5577425314
L(12)L(\frac12) ≈\approx 0.55774253140.5577425314
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+1.34T+2T2 1 + 1.34T + 2T^{2}
5 1+(1.26+2.19i)T+(−2.5+4.33i)T2 1 + (1.26 + 2.19i)T + (-2.5 + 4.33i)T^{2}
11 1+(−0.233+0.405i)T+(−5.5−9.52i)T2 1 + (-0.233 + 0.405i)T + (-5.5 - 9.52i)T^{2}
13 1+(−2.91+5.04i)T+(−6.5−11.2i)T2 1 + (-2.91 + 5.04i)T + (-6.5 - 11.2i)T^{2}
17 1+(1.93+3.35i)T+(−8.5+14.7i)T2 1 + (1.93 + 3.35i)T + (-8.5 + 14.7i)T^{2}
19 1+(1.09−1.89i)T+(−9.5−16.4i)T2 1 + (1.09 - 1.89i)T + (-9.5 - 16.4i)T^{2}
23 1+(0.0530+0.0918i)T+(−11.5+19.9i)T2 1 + (0.0530 + 0.0918i)T + (-11.5 + 19.9i)T^{2}
29 1+(−4.39−7.60i)T+(−14.5+25.1i)T2 1 + (-4.39 - 7.60i)T + (-14.5 + 25.1i)T^{2}
31 1−7.68T+31T2 1 - 7.68T + 31T^{2}
37 1+(−3.84+6.65i)T+(−18.5−32.0i)T2 1 + (-3.84 + 6.65i)T + (-18.5 - 32.0i)T^{2}
41 1+(−1.11+1.92i)T+(−20.5−35.5i)T2 1 + (-1.11 + 1.92i)T + (-20.5 - 35.5i)T^{2}
43 1+(0.613+1.06i)T+(−21.5+37.2i)T2 1 + (0.613 + 1.06i)T + (-21.5 + 37.2i)T^{2}
47 1+5.33T+47T2 1 + 5.33T + 47T^{2}
53 1+(0.358+0.620i)T+(−26.5+45.8i)T2 1 + (0.358 + 0.620i)T + (-26.5 + 45.8i)T^{2}
59 1−0.736T+59T2 1 - 0.736T + 59T^{2}
61 1+0.958T+61T2 1 + 0.958T + 61T^{2}
67 1+9.63T+67T2 1 + 9.63T + 67T^{2}
71 1+13.2T+71T2 1 + 13.2T + 71T^{2}
73 1+(5.13+8.89i)T+(−36.5+63.2i)T2 1 + (5.13 + 8.89i)T + (-36.5 + 63.2i)T^{2}
79 1+12.6T+79T2 1 + 12.6T + 79T^{2}
83 1+(−1.36−2.36i)T+(−41.5+71.8i)T2 1 + (-1.36 - 2.36i)T + (-41.5 + 71.8i)T^{2}
89 1+(−4.05+7.02i)T+(−44.5−77.0i)T2 1 + (-4.05 + 7.02i)T + (-44.5 - 77.0i)T^{2}
97 1+(6.80+11.7i)T+(−48.5+84.0i)T2 1 + (6.80 + 11.7i)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

−9.005580565919529751353984203649, −8.596492005100514195341095089825, −7.999929334081640314493986843501, −7.21197716512932420660530394202, −5.97319273124277563215589473575, −4.93942135405625988814934000333, −4.27797714492239535705207081251, −3.04225518968553811983517111953, −1.30922976347914737338850964670, −0.38739031963627469359228648863, 1.38951345142649593344204921578, 2.71193740472108446055551987836, 4.06485363713169926026262677578, 4.54988742258254241330735056671, 6.29375945383900139638674719230, 6.72023385843624936075354553970, 7.72964585173547274398376533640, 8.390499396459553033444606303060, 9.071969107341512586225973865224, 9.974458686890385131106484205722

Graph of the ZZ-function along the critical line