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Properties

Label 2-8550-1.1-c1-0-27
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 − 4.22·7-s + 8-s + 5.13·11-s − 3.16·13-s − 4.22·14-s + 16-s − 6.48·17-s − 19-s + 5.13·22-s + 7.56·23-s − 3.16·26-s − 4.22·28-s − 0.832·29-s − 4.51·31-s + 32-s − 6.48·34-s − 0.137·37-s − 38-s + 11.6·41-s + 2.51·43-s + 5.13·44-s + 7.56·46-s − 5.96·47-s + 10.8·49-s − 3.16·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 1.59·7-s + 0.353·8-s + 1.54·11-s − 0.878·13-s − 1.12·14-s + 0.250·16-s − 1.57·17-s − 0.229·19-s + 1.09·22-s + 1.57·23-s − 0.621·26-s − 0.798·28-s − 0.154·29-s − 0.810·31-s + 0.176·32-s − 1.11·34-s − 0.0226·37-s − 0.162·38-s + 1.81·41-s + 0.382·43-s + 0.774·44-s + 1.11·46-s − 0.869·47-s + 1.55·49-s − 0.439·52-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 2.4419132582.441913258
L(12)L(\frac12) ≈\approx 2.4419132582.441913258
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+4.22T+7T2 1 + 4.22T + 7T^{2}
11 1−5.13T+11T2 1 - 5.13T + 11T^{2}
13 1+3.16T+13T2 1 + 3.16T + 13T^{2}
17 1+6.48T+17T2 1 + 6.48T + 17T^{2}
23 1−7.56T+23T2 1 - 7.56T + 23T^{2}
29 1+0.832T+29T2 1 + 0.832T + 29T^{2}
31 1+4.51T+31T2 1 + 4.51T + 31T^{2}
37 1+0.137T+37T2 1 + 0.137T + 37T^{2}
41 1−11.6T+41T2 1 - 11.6T + 41T^{2}
43 1−2.51T+43T2 1 - 2.51T + 43T^{2}
47 1+5.96T+47T2 1 + 5.96T + 47T^{2}
53 1−0.225T+53T2 1 - 0.225T + 53T^{2}
59 1+5.39T+59T2 1 + 5.39T + 59T^{2}
61 1−14.4T+61T2 1 - 14.4T + 61T^{2}
67 1−4.11T+67T2 1 - 4.11T + 67T^{2}
71 1+3.82T+71T2 1 + 3.82T + 71T^{2}
73 1+4.70T+73T2 1 + 4.70T + 73T^{2}
79 1−10.6T+79T2 1 - 10.6T + 79T^{2}
83 1+12.0T+83T2 1 + 12.0T + 83T^{2}
89 1−10T+89T2 1 - 10T + 89T^{2}
97 1+3.93T+97T2 1 + 3.93T + 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

−7.32099966240337576191712517378, −6.96605402924763937548298463606, −6.41540425603527620442676737521, −5.87756376913086678243832051690, −4.86142866212559333207919586867, −4.20519562391212201786812656703, −3.54173208619207850515771218072, −2.80944422190270651768728525085, −2.00711274011848064251231028654, −0.66202374538604611070695406826, 0.66202374538604611070695406826, 2.00711274011848064251231028654, 2.80944422190270651768728525085, 3.54173208619207850515771218072, 4.20519562391212201786812656703, 4.86142866212559333207919586867, 5.87756376913086678243832051690, 6.41540425603527620442676737521, 6.96605402924763937548298463606, 7.32099966240337576191712517378

Graph of the ZZ-function along the critical line