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Properties

Label 2-110670-1.1-c1-0-24
Degree 22
Conductor 110670110670
Sign 11
Analytic cond. 883.704883.704
Root an. cond. 29.727129.7271
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s + 5-s − 6-s − 7-s − 8-s + 9-s − 10-s + 6·11-s + 12-s − 2·13-s + 14-s + 15-s + 16-s − 17-s − 18-s + 2·19-s + 20-s − 21-s − 6·22-s + 4·23-s − 24-s + 25-s + 2·26-s + 27-s − 28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.80·11-s + 0.288·12-s − 0.554·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.458·19-s + 0.223·20-s − 0.218·21-s − 1.27·22-s + 0.834·23-s − 0.204·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s − 0.188·28-s + ⋯

Functional equation

Λ(s)=(110670s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 110670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(110670s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 110670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 110670110670    =    2⋅3⋅5⋅7⋅17⋅312 \cdot 3 \cdot 5 \cdot 7 \cdot 17 \cdot 31
Sign: 11
Analytic conductor: 883.704883.704
Root analytic conductor: 29.727129.7271
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 110670, ( :1/2), 1)(2,\ 110670,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.4441928823.444192882
L(12)L(\frac12) ≈\approx 3.4441928823.444192882
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+T 1 + T
3 1−T 1 - T
5 1−T 1 - T
7 1+T 1 + T
17 1+T 1 + T
31 1−T 1 - T
good11 1−6T+pT2 1 - 6 T + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
19 1−2T+pT2 1 - 2 T + p T^{2}
23 1−4T+pT2 1 - 4 T + p T^{2}
29 1+4T+pT2 1 + 4 T + p T^{2}
37 1−10T+pT2 1 - 10 T + p T^{2}
41 1−10T+pT2 1 - 10 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1−2T+pT2 1 - 2 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1−8T+pT2 1 - 8 T + p T^{2}
61 1−2T+pT2 1 - 2 T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1+6T+pT2 1 + 6 T + p T^{2}
73 1−6T+pT2 1 - 6 T + p T^{2}
79 1−10T+pT2 1 - 10 T + p T^{2}
83 1−6T+pT2 1 - 6 T + p T^{2}
89 1+14T+pT2 1 + 14 T + p T^{2}
97 1+10T+pT2 1 + 10 T + p 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

−13.82806960419972, −13.02162777386651, −12.74364839381720, −12.22626196823301, −11.52023883301771, −11.26591584903252, −10.69441918565526, −9.937761830176429, −9.567107968328143, −9.259264255911144, −8.956124573025672, −8.306147732593149, −7.593783146988493, −7.300943933922205, −6.635785386106976, −6.298308482594574, −5.714611742748897, −4.975799333324204, −4.201940030494370, −3.834356182424799, −3.050580042954719, −2.532295385736511, −1.928157770091376, −1.148915831093911, −0.7028588819627286, 0.7028588819627286, 1.148915831093911, 1.928157770091376, 2.532295385736511, 3.050580042954719, 3.834356182424799, 4.201940030494370, 4.975799333324204, 5.714611742748897, 6.298308482594574, 6.635785386106976, 7.300943933922205, 7.593783146988493, 8.306147732593149, 8.956124573025672, 9.259264255911144, 9.567107968328143, 9.937761830176429, 10.69441918565526, 11.26591584903252, 11.52023883301771, 12.22626196823301, 12.74364839381720, 13.02162777386651, 13.82806960419972

Graph of the ZZ-function along the critical line