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Properties

Label 2-249690-1.1-c1-0-11
Degree 22
Conductor 249690249690
Sign 11
Analytic cond. 1993.781993.78
Root an. cond. 44.651844.6518
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 + 4·11-s − 12-s + 2·13-s + 14-s + 15-s + 16-s + 2·17-s − 18-s + 4·19-s − 20-s + 21-s − 4·22-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 + 1.20·11-s − 0.288·12-s + 0.554·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.917·19-s − 0.223·20-s + 0.218·21-s − 0.852·22-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)=(249690s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 249690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(249690s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 249690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 249690249690    =    2⋅3⋅5⋅7⋅29⋅412 \cdot 3 \cdot 5 \cdot 7 \cdot 29 \cdot 41
Sign: 11
Analytic conductor: 1993.781993.78
Root analytic conductor: 44.651844.6518
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 249690, ( :1/2), 1)(2,\ 249690,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.4134105401.413410540
L(12)L(\frac12) ≈\approx 1.4134105401.413410540
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
29 1+T 1 + T
41 1−T 1 - T
good11 1−4T+pT2 1 - 4 T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1−2T+pT2 1 - 2 T + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
23 1−4T+pT2 1 - 4 T + p T^{2}
31 1+8T+pT2 1 + 8 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
43 1+8T+pT2 1 + 8 T + p T^{2}
47 1+12T+pT2 1 + 12 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1−12T+pT2 1 - 12 T + p T^{2}
61 1−6T+pT2 1 - 6 T + p T^{2}
67 1−12T+pT2 1 - 12 T + p T^{2}
71 1−4T+pT2 1 - 4 T + p T^{2}
73 1−10T+pT2 1 - 10 T + p T^{2}
79 1+16T+pT2 1 + 16 T + p T^{2}
83 1−12T+pT2 1 - 12 T + p T^{2}
89 1+6T+pT2 1 + 6 T + p T^{2}
97 1−2T+pT2 1 - 2 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

−12.65054732625733, −12.39496892887975, −11.66895858620162, −11.43560426331467, −11.13193669291218, −10.64888317983139, −9.787143681713511, −9.722363347472355, −9.291160459676276, −8.525902529967352, −8.344381787406108, −7.640730128253130, −7.077020844377407, −6.796132938125541, −6.382576114213188, −5.682466820584866, −5.313507027922444, −4.701527833636740, −3.950559432126578, −3.416575817682459, −3.253832596448907, −2.201430853156303, −1.541288740703335, −1.032689648972985, −0.4506876140972005, 0.4506876140972005, 1.032689648972985, 1.541288740703335, 2.201430853156303, 3.253832596448907, 3.416575817682459, 3.950559432126578, 4.701527833636740, 5.313507027922444, 5.682466820584866, 6.382576114213188, 6.796132938125541, 7.077020844377407, 7.640730128253130, 8.344381787406108, 8.525902529967352, 9.291160459676276, 9.722363347472355, 9.787143681713511, 10.64888317983139, 11.13193669291218, 11.43560426331467, 11.66895858620162, 12.39496892887975, 12.65054732625733

Graph of the ZZ-function along the critical line