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Properties

Label 2-8550-1.1-c1-0-50
Degree 22
Conductor 85508550
Sign 11
Analytic cond. 68.272068.2720
Root an. cond. 8.262698.26269
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 4.69·7-s + 8-s − 6.40·11-s − 1.06·13-s + 4.69·14-s + 16-s + 1.91·17-s − 19-s − 6.40·22-s + 1.79·23-s − 1.06·26-s + 4.69·28-s − 2.93·29-s − 5.55·31-s + 32-s + 1.91·34-s + 11.4·37-s − 38-s + 1.14·41-s + 3.55·43-s − 6.40·44-s + 1.79·46-s + 10.8·47-s + 15.0·49-s − 1.06·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 1.77·7-s + 0.353·8-s − 1.93·11-s − 0.295·13-s + 1.25·14-s + 0.250·16-s + 0.465·17-s − 0.229·19-s − 1.36·22-s + 0.374·23-s − 0.208·26-s + 0.887·28-s − 0.545·29-s − 0.997·31-s + 0.176·32-s + 0.328·34-s + 1.87·37-s − 0.162·38-s + 0.178·41-s + 0.542·43-s − 0.966·44-s + 0.264·46-s + 1.58·47-s + 2.14·49-s − 0.147·52-s + ⋯

Functional equation

Λ(s)=(8550s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(8550s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 85508550    =    2⋅32⋅52⋅192 \cdot 3^{2} \cdot 5^{2} \cdot 19
Sign: 11
Analytic conductor: 68.272068.2720
Root analytic conductor: 8.262698.26269
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 8550, ( :1/2), 1)(2,\ 8550,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.7955714633.795571463
L(12)L(\frac12) ≈\approx 3.7955714633.795571463
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
19 1+T 1 + T
good7 1−4.69T+7T2 1 - 4.69T + 7T^{2}
11 1+6.40T+11T2 1 + 6.40T + 11T^{2}
13 1+1.06T+13T2 1 + 1.06T + 13T^{2}
17 1−1.91T+17T2 1 - 1.91T + 17T^{2}
23 1−1.79T+23T2 1 - 1.79T + 23T^{2}
29 1+2.93T+29T2 1 + 2.93T + 29T^{2}
31 1+5.55T+31T2 1 + 5.55T + 31T^{2}
37 1−11.4T+37T2 1 - 11.4T + 37T^{2}
41 1−1.14T+41T2 1 - 1.14T + 41T^{2}
43 1−3.55T+43T2 1 - 3.55T + 43T^{2}
47 1−10.8T+47T2 1 - 10.8T + 47T^{2}
53 1+8.69T+53T2 1 + 8.69T + 53T^{2}
59 1−5.63T+59T2 1 - 5.63T + 59T^{2}
61 1+3.39T+61T2 1 + 3.39T + 61T^{2}
67 1−8.82T+67T2 1 - 8.82T + 67T^{2}
71 1−1.42T+71T2 1 - 1.42T + 71T^{2}
73 1−12.6T+73T2 1 - 12.6T + 73T^{2}
79 1+1.96T+79T2 1 + 1.96T + 79T^{2}
83 1−16.2T+83T2 1 - 16.2T + 83T^{2}
89 1−10T+89T2 1 - 10T + 89T^{2}
97 1−14.9T+97T2 1 - 14.9T + 97T^{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

−7.62533134698806928934619752650, −7.44210278199316324917238706120, −6.16848512766407757022816162584, −5.43550523823473992024385017535, −5.04740447966574022547986681665, −4.47884954130528751319258693914, −3.56655545436772841553862821170, −2.46843252328081942670239562208, −2.10658192690874479963155257130, −0.855546094780470228366999836326, 0.855546094780470228366999836326, 2.10658192690874479963155257130, 2.46843252328081942670239562208, 3.56655545436772841553862821170, 4.47884954130528751319258693914, 5.04740447966574022547986681665, 5.43550523823473992024385017535, 6.16848512766407757022816162584, 7.44210278199316324917238706120, 7.62533134698806928934619752650

Graph of the ZZ-function along the critical line