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Properties

Label 2-133-133.10-c1-0-10
Degree 22
Conductor 133133
Sign −0.611−0.791i-0.611 - 0.791i
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  + (−1.52 − 1.81i)2-s + (−0.439 − 2.49i)3-s + (−0.623 + 3.53i)4-s + (−1.04 + 0.184i)5-s + (−3.84 + 4.58i)6-s + (−0.591 − 2.57i)7-s + (3.26 − 1.88i)8-s + (−3.18 + 1.16i)9-s + (1.92 + 1.61i)10-s − 2.17·11-s + 9.08·12-s + (4.74 + 3.98i)13-s + (−3.77 + 4.99i)14-s + (0.917 + 2.52i)15-s + (−1.61 − 0.588i)16-s + (1.08 − 2.97i)17-s + ⋯
L(s)  = 1  + (−1.07 − 1.28i)2-s + (−0.253 − 1.43i)3-s + (−0.311 + 1.76i)4-s + (−0.467 + 0.0824i)5-s + (−1.56 + 1.87i)6-s + (−0.223 − 0.974i)7-s + (1.15 − 0.665i)8-s + (−1.06 + 0.386i)9-s + (0.607 + 0.510i)10-s − 0.656·11-s + 2.62·12-s + (1.31 + 1.10i)13-s + (−1.00 + 1.33i)14-s + (0.236 + 0.651i)15-s + (−0.404 − 0.147i)16-s + (0.262 − 0.720i)17-s + ⋯

Functional equation

Λ(s)=(133s/2 ΓC(s) L(s)=((−0.611−0.791i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.611 - 0.791i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(133s/2 ΓC(s+1/2) L(s)=((−0.611−0.791i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.611 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 133133    =    7⋅197 \cdot 19
Sign: −0.611−0.791i-0.611 - 0.791i
Analytic conductor: 1.062011.06201
Root analytic conductor: 1.030531.03053
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ133(10,⋅)\chi_{133} (10, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 133, ( :1/2), −0.611−0.791i)(2,\ 133,\ (\ :1/2),\ -0.611 - 0.791i)

Particular Values

L(1)L(1) ≈\approx 0.178087+0.362561i0.178087 + 0.362561i
L(12)L(\frac12) ≈\approx 0.178087+0.362561i0.178087 + 0.362561i
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+(0.591+2.57i)T 1 + (0.591 + 2.57i)T
19 1+(2.57+3.51i)T 1 + (2.57 + 3.51i)T
good2 1+(1.52+1.81i)T+(−0.347+1.96i)T2 1 + (1.52 + 1.81i)T + (-0.347 + 1.96i)T^{2}
3 1+(0.439+2.49i)T+(−2.81+1.02i)T2 1 + (0.439 + 2.49i)T + (-2.81 + 1.02i)T^{2}
5 1+(1.04−0.184i)T+(4.69−1.71i)T2 1 + (1.04 - 0.184i)T + (4.69 - 1.71i)T^{2}
11 1+2.17T+11T2 1 + 2.17T + 11T^{2}
13 1+(−4.74−3.98i)T+(2.25+12.8i)T2 1 + (-4.74 - 3.98i)T + (2.25 + 12.8i)T^{2}
17 1+(−1.08+2.97i)T+(−13.0−10.9i)T2 1 + (-1.08 + 2.97i)T + (-13.0 - 10.9i)T^{2}
23 1+(4.58+3.84i)T+(3.99+22.6i)T2 1 + (4.58 + 3.84i)T + (3.99 + 22.6i)T^{2}
29 1+(1.85+0.327i)T+(27.2+9.91i)T2 1 + (1.85 + 0.327i)T + (27.2 + 9.91i)T^{2}
31 1+(−0.545−0.945i)T+(−15.5+26.8i)T2 1 + (-0.545 - 0.945i)T + (-15.5 + 26.8i)T^{2}
37 1+(−6.26+3.61i)T+(18.5−32.0i)T2 1 + (-6.26 + 3.61i)T + (18.5 - 32.0i)T^{2}
41 1+(−4.49+3.77i)T+(7.11−40.3i)T2 1 + (-4.49 + 3.77i)T + (7.11 - 40.3i)T^{2}
43 1+(4.46+1.62i)T+(32.9+27.6i)T2 1 + (4.46 + 1.62i)T + (32.9 + 27.6i)T^{2}
47 1+(−1.63−4.48i)T+(−36.0+30.2i)T2 1 + (-1.63 - 4.48i)T + (-36.0 + 30.2i)T^{2}
53 1+(1.86+0.328i)T+(49.8+18.1i)T2 1 + (1.86 + 0.328i)T + (49.8 + 18.1i)T^{2}
59 1+(−9.40−3.42i)T+(45.1+37.9i)T2 1 + (-9.40 - 3.42i)T + (45.1 + 37.9i)T^{2}
61 1+(−6.67+7.95i)T+(−10.5−60.0i)T2 1 + (-6.67 + 7.95i)T + (-10.5 - 60.0i)T^{2}
67 1+(−9.19+10.9i)T+(−11.6−65.9i)T2 1 + (-9.19 + 10.9i)T + (-11.6 - 65.9i)T^{2}
71 1+(−3.22+8.87i)T+(−54.3−45.6i)T2 1 + (-3.22 + 8.87i)T + (-54.3 - 45.6i)T^{2}
73 1+(−3.68+0.648i)T+(68.5−24.9i)T2 1 + (-3.68 + 0.648i)T + (68.5 - 24.9i)T^{2}
79 1+(1.99−5.49i)T+(−60.5−50.7i)T2 1 + (1.99 - 5.49i)T + (-60.5 - 50.7i)T^{2}
83 1+(−8.54−4.93i)T+(41.5+71.8i)T2 1 + (-8.54 - 4.93i)T + (41.5 + 71.8i)T^{2}
89 1+(1.14−6.49i)T+(−83.6−30.4i)T2 1 + (1.14 - 6.49i)T + (-83.6 - 30.4i)T^{2}
97 1+(2.13+12.0i)T+(−91.1+33.1i)T2 1 + (2.13 + 12.0i)T + (-91.1 + 33.1i)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

−12.39753482781848467973095521780, −11.39044764605321757590170824751, −10.90051923011729591487613246808, −9.579499000262112365531195387059, −8.312821888951373586056730318014, −7.51773178210447787674590425405, −6.43795045213455444485276896242, −3.88729773565505927305122091253, −2.15381244638495666907614637675, −0.61152147060681640547688238642, 3.75037833464762693024362985515, 5.51339711492188987594581211275, 6.03072503249940143700816017122, 8.027175690369944214823895031200, 8.466862230928425251872017583149, 9.744115546772427045789734490221, 10.30187101214909844389931742984, 11.46357037280792581933053584087, 12.98495806458507061541735826527, 14.74574867841517582383671159273

Graph of the ZZ-function along the critical line