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Properties

Label 2-1323-63.58-c1-0-21
Degree 22
Conductor 13231323
Sign −0.703+0.711i-0.703 + 0.711i
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  − 0.670·2-s − 1.55·4-s + (−0.712 − 1.23i)5-s + 2.38·8-s + (0.477 + 0.827i)10-s + (−2.46 + 4.27i)11-s + (1.37 − 2.38i)13-s + 1.50·16-s + (0.559 + 0.969i)17-s + (2.00 − 3.47i)19-s + (1.10 + 1.91i)20-s + (1.65 − 2.86i)22-s + (2.71 + 4.70i)23-s + (1.48 − 2.57i)25-s + (−0.923 + 1.59i)26-s + ⋯
L(s)  = 1  − 0.473·2-s − 0.775·4-s + (−0.318 − 0.551i)5-s + 0.841·8-s + (0.151 + 0.261i)10-s + (−0.743 + 1.28i)11-s + (0.381 − 0.661i)13-s + 0.376·16-s + (0.135 + 0.235i)17-s + (0.460 − 0.797i)19-s + (0.247 + 0.427i)20-s + (0.352 − 0.610i)22-s + (0.566 + 0.981i)23-s + (0.296 − 0.514i)25-s + (−0.181 + 0.313i)26-s + ⋯

Functional equation

Λ(s)=(1323s/2 ΓC(s) L(s)=((−0.703+0.711i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.703 + 0.711i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1323s/2 ΓC(s+1/2) L(s)=((−0.703+0.711i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.703 + 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13231323    =    33⋅723^{3} \cdot 7^{2}
Sign: −0.703+0.711i-0.703 + 0.711i
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), −0.703+0.711i)(2,\ 1323,\ (\ :1/2),\ -0.703 + 0.711i)

Particular Values

L(1)L(1) ≈\approx 0.42601494440.4260149444
L(12)L(\frac12) ≈\approx 0.42601494440.4260149444
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+0.670T+2T2 1 + 0.670T + 2T^{2}
5 1+(0.712+1.23i)T+(−2.5+4.33i)T2 1 + (0.712 + 1.23i)T + (-2.5 + 4.33i)T^{2}
11 1+(2.46−4.27i)T+(−5.5−9.52i)T2 1 + (2.46 - 4.27i)T + (-5.5 - 9.52i)T^{2}
13 1+(−1.37+2.38i)T+(−6.5−11.2i)T2 1 + (-1.37 + 2.38i)T + (-6.5 - 11.2i)T^{2}
17 1+(−0.559−0.969i)T+(−8.5+14.7i)T2 1 + (-0.559 - 0.969i)T + (-8.5 + 14.7i)T^{2}
19 1+(−2.00+3.47i)T+(−9.5−16.4i)T2 1 + (-2.00 + 3.47i)T + (-9.5 - 16.4i)T^{2}
23 1+(−2.71−4.70i)T+(−11.5+19.9i)T2 1 + (-2.71 - 4.70i)T + (-11.5 + 19.9i)T^{2}
29 1+(3.40+5.89i)T+(−14.5+25.1i)T2 1 + (3.40 + 5.89i)T + (-14.5 + 25.1i)T^{2}
31 1+2.50T+31T2 1 + 2.50T + 31T^{2}
37 1+(−0.709+1.22i)T+(−18.5−32.0i)T2 1 + (-0.709 + 1.22i)T + (-18.5 - 32.0i)T^{2}
41 1+(−0.124+0.215i)T+(−20.5−35.5i)T2 1 + (-0.124 + 0.215i)T + (-20.5 - 35.5i)T^{2}
43 1+(0.498+0.863i)T+(−21.5+37.2i)T2 1 + (0.498 + 0.863i)T + (-21.5 + 37.2i)T^{2}
47 1+9.47T+47T2 1 + 9.47T + 47T^{2}
53 1+(−0.410−0.710i)T+(−26.5+45.8i)T2 1 + (-0.410 - 0.710i)T + (-26.5 + 45.8i)T^{2}
59 1+6.58T+59T2 1 + 6.58T + 59T^{2}
61 1+0.0752T+61T2 1 + 0.0752T + 61T^{2}
67 1+12.5T+67T2 1 + 12.5T + 67T^{2}
71 1+0.0804T+71T2 1 + 0.0804T + 71T^{2}
73 1+(5.34+9.25i)T+(−36.5+63.2i)T2 1 + (5.34 + 9.25i)T + (-36.5 + 63.2i)T^{2}
79 1+1.84T+79T2 1 + 1.84T + 79T^{2}
83 1+(7.23+12.5i)T+(−41.5+71.8i)T2 1 + (7.23 + 12.5i)T + (-41.5 + 71.8i)T^{2}
89 1+(−6.76+11.7i)T+(−44.5−77.0i)T2 1 + (-6.76 + 11.7i)T + (-44.5 - 77.0i)T^{2}
97 1+(2.70+4.67i)T+(−48.5+84.0i)T2 1 + (2.70 + 4.67i)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

−9.326933627045154904360722795316, −8.578298131436463693575028774695, −7.73371884616265091431299103727, −7.31079288103065972813610652947, −5.82584337572000479141232726822, −4.93565499795251242310140856933, −4.37912342705926221382120839713, −3.16409609697117601197940798274, −1.63187891132152316254784359013, −0.23956791994584263408497247871, 1.28641097730475574372882863875, 3.01139614429900924605184310798, 3.75673115570171331355476293732, 4.91578174256287537748040823768, 5.71866625690005212665657410139, 6.81096157573518790129136001415, 7.68209673046353272611347637171, 8.405358313918884171910345858901, 9.026597991910496189643622770024, 9.855026110699888530405367452531

Graph of the ZZ-function along the critical line