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Properties

Label 2-133-133.103-c1-0-10
Degree 22
Conductor 133133
Sign 0.621+0.783i0.621 + 0.783i
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  + (2.11 − 1.21i)2-s + 0.519·3-s + (1.96 − 3.40i)4-s + (−2.56 + 1.47i)5-s + (1.09 − 0.632i)6-s + (0.877 + 2.49i)7-s − 4.71i·8-s − 2.73·9-s + (−3.60 + 6.24i)10-s + (−2.13 − 3.69i)11-s + (1.02 − 1.77i)12-s + (0.490 + 0.848i)13-s + (4.89 + 4.19i)14-s + (−1.33 + 0.768i)15-s + (−1.81 − 3.13i)16-s − 4.99i·17-s + ⋯
L(s)  = 1  + (1.49 − 0.861i)2-s + 0.299·3-s + (0.984 − 1.70i)4-s + (−1.14 + 0.661i)5-s + (0.447 − 0.258i)6-s + (0.331 + 0.943i)7-s − 1.66i·8-s − 0.910·9-s + (−1.13 + 1.97i)10-s + (−0.642 − 1.11i)11-s + (0.295 − 0.511i)12-s + (0.135 + 0.235i)13-s + (1.30 + 1.12i)14-s + (−0.343 + 0.198i)15-s + (−0.452 − 0.784i)16-s − 1.21i·17-s + ⋯

Functional equation

Λ(s)=(133s/2 ΓC(s) L(s)=((0.621+0.783i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 + 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(133s/2 ΓC(s+1/2) L(s)=((0.621+0.783i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.621 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 133133    =    7⋅197 \cdot 19
Sign: 0.621+0.783i0.621 + 0.783i
Analytic conductor: 1.062011.06201
Root analytic conductor: 1.030531.03053
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ133(103,⋅)\chi_{133} (103, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 133, ( :1/2), 0.621+0.783i)(2,\ 133,\ (\ :1/2),\ 0.621 + 0.783i)

Particular Values

L(1)L(1) ≈\approx 1.83748−0.888392i1.83748 - 0.888392i
L(12)L(\frac12) ≈\approx 1.83748−0.888392i1.83748 - 0.888392i
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+(−0.877−2.49i)T 1 + (-0.877 - 2.49i)T
19 1+(−1.53−4.07i)T 1 + (-1.53 - 4.07i)T
good2 1+(−2.11+1.21i)T+(1−1.73i)T2 1 + (-2.11 + 1.21i)T + (1 - 1.73i)T^{2}
3 1−0.519T+3T2 1 - 0.519T + 3T^{2}
5 1+(2.56−1.47i)T+(2.5−4.33i)T2 1 + (2.56 - 1.47i)T + (2.5 - 4.33i)T^{2}
11 1+(2.13+3.69i)T+(−5.5+9.52i)T2 1 + (2.13 + 3.69i)T + (-5.5 + 9.52i)T^{2}
13 1+(−0.490−0.848i)T+(−6.5+11.2i)T2 1 + (-0.490 - 0.848i)T + (-6.5 + 11.2i)T^{2}
17 1+4.99iT−17T2 1 + 4.99iT - 17T^{2}
23 1−7.91T+23T2 1 - 7.91T + 23T^{2}
29 1+(−1.16+0.674i)T+(14.5−25.1i)T2 1 + (-1.16 + 0.674i)T + (14.5 - 25.1i)T^{2}
31 1+(2.73+4.74i)T+(−15.5+26.8i)T2 1 + (2.73 + 4.74i)T + (-15.5 + 26.8i)T^{2}
37 1+(1.05+0.611i)T+(18.5+32.0i)T2 1 + (1.05 + 0.611i)T + (18.5 + 32.0i)T^{2}
41 1+(1.34−2.32i)T+(−20.5−35.5i)T2 1 + (1.34 - 2.32i)T + (-20.5 - 35.5i)T^{2}
43 1+(2.72−4.71i)T+(−21.5−37.2i)T2 1 + (2.72 - 4.71i)T + (-21.5 - 37.2i)T^{2}
47 1+4.39iT−47T2 1 + 4.39iT - 47T^{2}
53 1+(−6.59−3.80i)T+(26.5+45.8i)T2 1 + (-6.59 - 3.80i)T + (26.5 + 45.8i)T^{2}
59 1+9.34T+59T2 1 + 9.34T + 59T^{2}
61 1−3.37iT−61T2 1 - 3.37iT - 61T^{2}
67 1+(1.50+0.869i)T+(33.5+58.0i)T2 1 + (1.50 + 0.869i)T + (33.5 + 58.0i)T^{2}
71 1+(12.4+7.17i)T+(35.5+61.4i)T2 1 + (12.4 + 7.17i)T + (35.5 + 61.4i)T^{2}
73 1+4.63iT−73T2 1 + 4.63iT - 73T^{2}
79 1+(4.21−2.43i)T+(39.5−68.4i)T2 1 + (4.21 - 2.43i)T + (39.5 - 68.4i)T^{2}
83 1−2.39iT−83T2 1 - 2.39iT - 83T^{2}
89 1−14.4T+89T2 1 - 14.4T + 89T^{2}
97 1+(−7.58+13.1i)T+(−48.5−84.0i)T2 1 + (-7.58 + 13.1i)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

−13.16279738709227281188563298161, −11.77532108680217558746941161353, −11.57123719974851647880254276002, −10.72376649981835694893469364297, −8.920453012703103530070283315669, −7.70680118560682091892734244170, −6.01353764380281197249247243635, −5.01702300190554006196795664577, −3.39181981719551954086264329414, −2.75980639365218901105545116466, 3.29086864142919688578784963342, 4.44021427449458626676963438529, 5.23796882008971385966191194060, 6.96505531628453294274001017474, 7.71899527203791971353909172222, 8.674175227410196203293061683483, 10.71431735255911778651490372987, 11.80219744449588043803193458302, 12.76018169870003352725898599517, 13.39090690322505918111048333238

Graph of the ZZ-function along the critical line