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Properties

Label 2-1332-37.12-c1-0-3
Degree 22
Conductor 13321332
Sign −0.429−0.903i-0.429 - 0.903i
Analytic cond. 10.636010.6360
Root an. cond. 3.261293.26129
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  + (0.245 − 0.0892i)5-s + (−2.06 + 0.750i)7-s + (−2.40 − 4.15i)11-s + (−0.355 + 2.01i)13-s + (−0.213 − 1.21i)17-s + (5.98 + 5.02i)19-s + (−2.71 + 4.70i)23-s + (−3.77 + 3.17i)25-s + (2.10 + 3.64i)29-s − 2.24·31-s + (−0.438 + 0.367i)35-s + (5.68 + 2.15i)37-s + (−1.39 + 7.88i)41-s − 8.68·43-s + (−3.17 + 5.50i)47-s + ⋯
L(s)  = 1  + (0.109 − 0.0399i)5-s + (−0.778 + 0.283i)7-s + (−0.723 − 1.25i)11-s + (−0.0987 + 0.559i)13-s + (−0.0518 − 0.294i)17-s + (1.37 + 1.15i)19-s + (−0.566 + 0.982i)23-s + (−0.755 + 0.634i)25-s + (0.390 + 0.676i)29-s − 0.402·31-s + (−0.0740 + 0.0621i)35-s + (0.934 + 0.355i)37-s + (−0.217 + 1.23i)41-s − 1.32·43-s + (−0.463 + 0.802i)47-s + ⋯

Functional equation

Λ(s)=(1332s/2 ΓC(s) L(s)=((−0.429−0.903i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.429 - 0.903i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1332s/2 ΓC(s+1/2) L(s)=((−0.429−0.903i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.429 - 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13321332    =    22⋅32⋅372^{2} \cdot 3^{2} \cdot 37
Sign: −0.429−0.903i-0.429 - 0.903i
Analytic conductor: 10.636010.6360
Root analytic conductor: 3.261293.26129
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1332(937,⋅)\chi_{1332} (937, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1332, ( :1/2), −0.429−0.903i)(2,\ 1332,\ (\ :1/2),\ -0.429 - 0.903i)

Particular Values

L(1)L(1) ≈\approx 0.79220269120.7922026912
L(12)L(\frac12) ≈\approx 0.79220269120.7922026912
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)
bad2 1 1
3 1 1
37 1+(−5.68−2.15i)T 1 + (-5.68 - 2.15i)T
good5 1+(−0.245+0.0892i)T+(3.83−3.21i)T2 1 + (-0.245 + 0.0892i)T + (3.83 - 3.21i)T^{2}
7 1+(2.06−0.750i)T+(5.36−4.49i)T2 1 + (2.06 - 0.750i)T + (5.36 - 4.49i)T^{2}
11 1+(2.40+4.15i)T+(−5.5+9.52i)T2 1 + (2.40 + 4.15i)T + (-5.5 + 9.52i)T^{2}
13 1+(0.355−2.01i)T+(−12.2−4.44i)T2 1 + (0.355 - 2.01i)T + (-12.2 - 4.44i)T^{2}
17 1+(0.213+1.21i)T+(−15.9+5.81i)T2 1 + (0.213 + 1.21i)T + (-15.9 + 5.81i)T^{2}
19 1+(−5.98−5.02i)T+(3.29+18.7i)T2 1 + (-5.98 - 5.02i)T + (3.29 + 18.7i)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+(−2.10−3.64i)T+(−14.5+25.1i)T2 1 + (-2.10 - 3.64i)T + (-14.5 + 25.1i)T^{2}
31 1+2.24T+31T2 1 + 2.24T + 31T^{2}
41 1+(1.39−7.88i)T+(−38.5−14.0i)T2 1 + (1.39 - 7.88i)T + (-38.5 - 14.0i)T^{2}
43 1+8.68T+43T2 1 + 8.68T + 43T^{2}
47 1+(3.17−5.50i)T+(−23.5−40.7i)T2 1 + (3.17 - 5.50i)T + (-23.5 - 40.7i)T^{2}
53 1+(−5.14−1.87i)T+(40.6+34.0i)T2 1 + (-5.14 - 1.87i)T + (40.6 + 34.0i)T^{2}
59 1+(13.9+5.06i)T+(45.1+37.9i)T2 1 + (13.9 + 5.06i)T + (45.1 + 37.9i)T^{2}
61 1+(−0.582+3.30i)T+(−57.3−20.8i)T2 1 + (-0.582 + 3.30i)T + (-57.3 - 20.8i)T^{2}
67 1+(−1.62+0.590i)T+(51.3−43.0i)T2 1 + (-1.62 + 0.590i)T + (51.3 - 43.0i)T^{2}
71 1+(5.88+4.94i)T+(12.3+69.9i)T2 1 + (5.88 + 4.94i)T + (12.3 + 69.9i)T^{2}
73 1+8.22T+73T2 1 + 8.22T + 73T^{2}
79 1+(−10.8+3.94i)T+(60.5−50.7i)T2 1 + (-10.8 + 3.94i)T + (60.5 - 50.7i)T^{2}
83 1+(−1.67−9.47i)T+(−77.9+28.3i)T2 1 + (-1.67 - 9.47i)T + (-77.9 + 28.3i)T^{2}
89 1+(−8.92−3.24i)T+(68.1+57.2i)T2 1 + (-8.92 - 3.24i)T + (68.1 + 57.2i)T^{2}
97 1+(−0.412+0.714i)T+(−48.5−84.0i)T2 1 + (-0.412 + 0.714i)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

−9.630723987675784516024403680427, −9.357156263225476926360351754304, −8.129868499802952711503014690317, −7.62597490414699292654624612626, −6.41158057617828638309141933559, −5.79913904651095778360279705648, −4.98804648169354886569691801492, −3.53975307619953557394878145003, −3.01663288763618395481253948157, −1.48054664589223682256735506907, 0.32061277125877477902542403863, 2.14800535826854608563126703548, 3.07768681258751882312028658638, 4.24778027663606379623119927766, 5.11609196370110838189674986520, 6.07776218080445578291510978448, 7.01191691676954372492851133515, 7.60738342602784988577145159030, 8.538733172901150758552483424500, 9.646525584425829443123796524696

Graph of the ZZ-function along the critical line