↑

Properties

Label 2-286650-1.1-c1-0-10
Degree 22
Conductor 286650286650
Sign 11
Analytic cond. 2288.912288.91
Root an. cond. 47.842547.8425
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 8-s + 11-s − 13-s + 16-s − 8·17-s + 4·19-s − 22-s + 4·23-s + 26-s + 9·29-s − 8·31-s − 32-s + 8·34-s + 2·37-s − 4·38-s + 5·41-s − 5·43-s + 44-s − 4·46-s − 4·47-s − 52-s + 9·53-s − 9·58-s − 15·59-s − 10·61-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s + 0.301·11-s − 0.277·13-s + 1/4·16-s − 1.94·17-s + 0.917·19-s − 0.213·22-s + 0.834·23-s + 0.196·26-s + 1.67·29-s − 1.43·31-s − 0.176·32-s + 1.37·34-s + 0.328·37-s − 0.648·38-s + 0.780·41-s − 0.762·43-s + 0.150·44-s − 0.589·46-s − 0.583·47-s − 0.138·52-s + 1.23·53-s − 1.18·58-s − 1.95·59-s − 1.28·61-s + ⋯

Functional equation

Λ(s)=(286650s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(286650s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 286650286650    =    2⋅32⋅52⋅72⋅132 \cdot 3^{2} \cdot 5^{2} \cdot 7^{2} \cdot 13
Sign: 11
Analytic conductor: 2288.912288.91
Root analytic conductor: 47.842547.8425
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 286650, ( :1/2), 1)(2,\ 286650,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.44260628720.4426062872
L(12)L(\frac12) ≈\approx 0.44260628720.4426062872
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 1
5 1 1
7 1 1
13 1+T 1 + T
good11 1−T+pT2 1 - T + p T^{2}
17 1+8T+pT2 1 + 8 T + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
23 1−4T+pT2 1 - 4 T + p T^{2}
29 1−9T+pT2 1 - 9 T + p T^{2}
31 1+8T+pT2 1 + 8 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1−5T+pT2 1 - 5 T + p T^{2}
43 1+5T+pT2 1 + 5 T + p T^{2}
47 1+4T+pT2 1 + 4 T + p T^{2}
53 1−9T+pT2 1 - 9 T + p T^{2}
59 1+15T+pT2 1 + 15 T + p T^{2}
61 1+10T+pT2 1 + 10 T + p T^{2}
67 1+pT2 1 + p T^{2}
71 1+10T+pT2 1 + 10 T + p T^{2}
73 1+8T+pT2 1 + 8 T + p T^{2}
79 1+12T+pT2 1 + 12 T + p T^{2}
83 1+7T+pT2 1 + 7 T + p T^{2}
89 1+15T+pT2 1 + 15 T + p T^{2}
97 1−10T+pT2 1 - 10 T + p T^{2}
show more
show less
   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

−12.62028767347004, −12.18841871410763, −11.74284074954532, −11.22925757163876, −10.89720051462516, −10.49753438082362, −9.883601662808675, −9.434935295652811, −9.003215649327005, −8.697185858469691, −8.191647310108367, −7.529971100896754, −7.125040520501228, −6.786231653280276, −6.237904663688475, −5.720659885953348, −5.114486862719981, −4.469785681393737, −4.230775826034911, −3.305337627781935, −2.820383573734393, −2.422295400807566, −1.505732056868602, −1.267424448233722, −0.2013223196167277, 0.2013223196167277, 1.267424448233722, 1.505732056868602, 2.422295400807566, 2.820383573734393, 3.305337627781935, 4.230775826034911, 4.469785681393737, 5.114486862719981, 5.720659885953348, 6.237904663688475, 6.786231653280276, 7.125040520501228, 7.529971100896754, 8.191647310108367, 8.697185858469691, 9.003215649327005, 9.434935295652811, 9.883601662808675, 10.49753438082362, 10.89720051462516, 11.22925757163876, 11.74284074954532, 12.18841871410763, 12.62028767347004

Graph of the ZZ-function along the critical line