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Properties

Label 2-133-133.102-c1-0-3
Degree 22
Conductor 133133
Sign 0.958+0.284i0.958 + 0.284i
Analytic cond. 1.062011.06201
Root an. cond. 1.030531.03053
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  + (0.143 − 0.248i)2-s − 3.06·3-s + (0.958 + 1.66i)4-s + (1.79 − 3.10i)5-s + (−0.439 + 0.760i)6-s + (2.10 + 1.60i)7-s + 1.12·8-s + 6.39·9-s + (−0.513 − 0.888i)10-s + (1.39 − 2.41i)11-s + (−2.93 − 5.09i)12-s + (0.705 − 1.22i)13-s + (0.700 − 0.291i)14-s + (−5.48 + 9.50i)15-s + (−1.75 + 3.04i)16-s − 2.10·17-s + ⋯
L(s)  = 1  + (0.101 − 0.175i)2-s − 1.76·3-s + (0.479 + 0.830i)4-s + (0.800 − 1.38i)5-s + (−0.179 + 0.310i)6-s + (0.794 + 0.607i)7-s + 0.397·8-s + 2.13·9-s + (−0.162 − 0.281i)10-s + (0.420 − 0.727i)11-s + (−0.848 − 1.46i)12-s + (0.195 − 0.338i)13-s + (0.187 − 0.0779i)14-s + (−1.41 + 2.45i)15-s + (−0.439 + 0.760i)16-s − 0.509·17-s + ⋯

Functional equation

Λ(s)=(133s/2 ΓC(s) L(s)=((0.958+0.284i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.284i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(133s/2 ΓC(s+1/2) L(s)=((0.958+0.284i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 + 0.284i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 133133    =    7⋅197 \cdot 19
Sign: 0.958+0.284i0.958 + 0.284i
Analytic conductor: 1.062011.06201
Root analytic conductor: 1.030531.03053
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ133(102,⋅)\chi_{133} (102, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 133, ( :1/2), 0.958+0.284i)(2,\ 133,\ (\ :1/2),\ 0.958 + 0.284i)

Particular Values

L(1)L(1) ≈\approx 0.920427−0.133511i0.920427 - 0.133511i
L(12)L(\frac12) ≈\approx 0.920427−0.133511i0.920427 - 0.133511i
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)
bad7 1+(−2.10−1.60i)T 1 + (-2.10 - 1.60i)T
19 1+(−2.60−3.49i)T 1 + (-2.60 - 3.49i)T
good2 1+(−0.143+0.248i)T+(−1−1.73i)T2 1 + (-0.143 + 0.248i)T + (-1 - 1.73i)T^{2}
3 1+3.06T+3T2 1 + 3.06T + 3T^{2}
5 1+(−1.79+3.10i)T+(−2.5−4.33i)T2 1 + (-1.79 + 3.10i)T + (-2.5 - 4.33i)T^{2}
11 1+(−1.39+2.41i)T+(−5.5−9.52i)T2 1 + (-1.39 + 2.41i)T + (-5.5 - 9.52i)T^{2}
13 1+(−0.705+1.22i)T+(−6.5−11.2i)T2 1 + (-0.705 + 1.22i)T + (-6.5 - 11.2i)T^{2}
17 1+2.10T+17T2 1 + 2.10T + 17T^{2}
23 1+4.91T+23T2 1 + 4.91T + 23T^{2}
29 1+(−0.941+1.63i)T+(−14.5−25.1i)T2 1 + (-0.941 + 1.63i)T + (-14.5 - 25.1i)T^{2}
31 1+(0.474−0.822i)T+(−15.5−26.8i)T2 1 + (0.474 - 0.822i)T + (-15.5 - 26.8i)T^{2}
37 1+(3.59+6.23i)T+(−18.5+32.0i)T2 1 + (3.59 + 6.23i)T + (-18.5 + 32.0i)T^{2}
41 1+(−1.31−2.27i)T+(−20.5+35.5i)T2 1 + (-1.31 - 2.27i)T + (-20.5 + 35.5i)T^{2}
43 1+(−0.579−1.00i)T+(−21.5+37.2i)T2 1 + (-0.579 - 1.00i)T + (-21.5 + 37.2i)T^{2}
47 1+7.78T+47T2 1 + 7.78T + 47T^{2}
53 1+(−6.77−11.7i)T+(−26.5+45.8i)T2 1 + (-6.77 - 11.7i)T + (-26.5 + 45.8i)T^{2}
59 1+2.99T+59T2 1 + 2.99T + 59T^{2}
61 1+4.13T+61T2 1 + 4.13T + 61T^{2}
67 1+(5.42+9.39i)T+(−33.5+58.0i)T2 1 + (5.42 + 9.39i)T + (-33.5 + 58.0i)T^{2}
71 1+(0.00205+0.00355i)T+(−35.5+61.4i)T2 1 + (0.00205 + 0.00355i)T + (-35.5 + 61.4i)T^{2}
73 1+5.65T+73T2 1 + 5.65T + 73T^{2}
79 1+(0.322−0.559i)T+(−39.5−68.4i)T2 1 + (0.322 - 0.559i)T + (-39.5 - 68.4i)T^{2}
83 1+4.44T+83T2 1 + 4.44T + 83T^{2}
89 1+6.57T+89T2 1 + 6.57T + 89T^{2}
97 1+(−2.26−3.91i)T+(−48.5+84.0i)T2 1 + (-2.26 - 3.91i)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

−12.74990965157924068808909586968, −12.10992156685468570051692405608, −11.50763098783882840420615302626, −10.47847040143564323662563258787, −9.029999511194116280986244074105, −7.88012151193147657256718747482, −6.18983241989298958897813525420, −5.49147390106762532945132093511, −4.35907347511661554483186196866, −1.55144154159546523126369384269, 1.74179882755667794595420274681, 4.61747598304907069939802656591, 5.71588318412209103097100204403, 6.71958497425598866327737939128, 7.10436038939468253846017820449, 9.867451695366802523007048819617, 10.41383191719442953178756380455, 11.23745847813281092447714269995, 11.75032680002974128226520775564, 13.48243211248802222085069878654

Graph of the ZZ-function along the critical line