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Properties

Label 2-110670-1.1-c1-0-25
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 + 6·13-s + 14-s − 15-s + 16-s − 17-s − 18-s + 6·19-s + 20-s + 21-s − 6·22-s + 6·23-s + 24-s + 25-s − 6·26-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 + 1.66·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 1.37·19-s + 0.223·20-s + 0.218·21-s − 1.27·22-s + 1.25·23-s + 0.204·24-s + 1/5·25-s − 1.17·26-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 2.7575173032.757517303
L(12)L(\frac12) ≈\approx 2.7575173032.757517303
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 1−6T+pT2 1 - 6 T + p T^{2}
19 1−6T+pT2 1 - 6 T + p T^{2}
23 1−6T+pT2 1 - 6 T + p T^{2}
29 1−2T+pT2 1 - 2 T + p T^{2}
37 1−8T+pT2 1 - 8 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1−6T+pT2 1 - 6 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1+pT2 1 + p T^{2}
59 1+pT2 1 + p T^{2}
61 1−10T+pT2 1 - 10 T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1+4T+pT2 1 + 4 T + p T^{2}
73 1−6T+pT2 1 - 6 T + p T^{2}
79 1−4T+pT2 1 - 4 T + p T^{2}
83 1+4T+pT2 1 + 4 T + p T^{2}
89 1+2T+pT2 1 + 2 T + p T^{2}
97 1−12T+pT2 1 - 12 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

−13.63562567437381, −13.14369320461835, −12.65465512406251, −11.98563709599159, −11.57737137783467, −11.22767361546962, −10.79667006222685, −10.20322112840294, −9.560175047424054, −9.280085123860994, −8.909361679879732, −8.317777397215251, −7.682654035745839, −6.891230626099738, −6.687549208637810, −6.254709539567245, −5.660394531703729, −5.203832136699348, −4.291982342468172, −3.786466710391645, −3.252449502205915, −2.541221805201484, −1.537293524268390, −1.147271639249301, −0.7285598703238091, 0.7285598703238091, 1.147271639249301, 1.537293524268390, 2.541221805201484, 3.252449502205915, 3.786466710391645, 4.291982342468172, 5.203832136699348, 5.660394531703729, 6.254709539567245, 6.687549208637810, 6.891230626099738, 7.682654035745839, 8.317777397215251, 8.909361679879732, 9.280085123860994, 9.560175047424054, 10.20322112840294, 10.79667006222685, 11.22767361546962, 11.57737137783467, 11.98563709599159, 12.65465512406251, 13.14369320461835, 13.63562567437381

Graph of the ZZ-function along the critical line