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Properties

Label 2-8550-1.1-c1-0-133
Degree 22
Conductor 85508550
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 1.03·7-s + 8-s + 3.86·11-s + 1.03·13-s − 1.03·14-s + 16-s − 7.46·17-s − 19-s + 3.86·22-s + 1.46·23-s + 1.03·26-s − 1.03·28-s − 9.52·29-s + 2.92·31-s + 32-s − 7.46·34-s − 6.69·37-s − 38-s − 6.69·41-s − 1.79·43-s + 3.86·44-s + 1.46·46-s − 9.46·47-s − 5.92·49-s + 1.03·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 0.391·7-s + 0.353·8-s + 1.16·11-s + 0.287·13-s − 0.276·14-s + 0.250·16-s − 1.81·17-s − 0.229·19-s + 0.823·22-s + 0.305·23-s + 0.203·26-s − 0.195·28-s − 1.76·29-s + 0.525·31-s + 0.176·32-s − 1.28·34-s − 1.10·37-s − 0.162·38-s − 1.04·41-s − 0.273·43-s + 0.582·44-s + 0.215·46-s − 1.38·47-s − 0.846·49-s + 0.143·52-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: −1-1
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: 11
Selberg data: (2, 8550, ( :1/2), −1)(2,\ 8550,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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+1.03T+7T2 1 + 1.03T + 7T^{2}
11 1−3.86T+11T2 1 - 3.86T + 11T^{2}
13 1−1.03T+13T2 1 - 1.03T + 13T^{2}
17 1+7.46T+17T2 1 + 7.46T + 17T^{2}
23 1−1.46T+23T2 1 - 1.46T + 23T^{2}
29 1+9.52T+29T2 1 + 9.52T + 29T^{2}
31 1−2.92T+31T2 1 - 2.92T + 31T^{2}
37 1+6.69T+37T2 1 + 6.69T + 37T^{2}
41 1+6.69T+41T2 1 + 6.69T + 41T^{2}
43 1+1.79T+43T2 1 + 1.79T + 43T^{2}
47 1+9.46T+47T2 1 + 9.46T + 47T^{2}
53 1+6T+53T2 1 + 6T + 53T^{2}
59 1−12.6T+59T2 1 - 12.6T + 59T^{2}
61 1+2T+61T2 1 + 2T + 61T^{2}
67 1−3.58T+67T2 1 - 3.58T + 67T^{2}
71 1−15.4T+71T2 1 - 15.4T + 71T^{2}
73 1+13.3T+73T2 1 + 13.3T + 73T^{2}
79 1+4T+79T2 1 + 4T + 79T^{2}
83 1−4.39T+83T2 1 - 4.39T + 83T^{2}
89 1+1.03T+89T2 1 + 1.03T + 89T^{2}
97 1−17.2T+97T2 1 - 17.2T + 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

−7.06803821910788384113658907932, −6.67045166744698232027563618257, −6.21144673719911554374151535173, −5.26636700491076218577964732557, −4.59711051783282614574722854794, −3.81895567213535687052661989691, −3.33949799656411431151792646017, −2.21257853026140678874406563119, −1.52380750299865813559141647063, 0, 1.52380750299865813559141647063, 2.21257853026140678874406563119, 3.33949799656411431151792646017, 3.81895567213535687052661989691, 4.59711051783282614574722854794, 5.26636700491076218577964732557, 6.21144673719911554374151535173, 6.67045166744698232027563618257, 7.06803821910788384113658907932

Graph of the ZZ-function along the critical line