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Properties

Label 2-4788-1.1-c1-0-12
Degree 22
Conductor 47884788
Sign 11
Analytic cond. 38.232338.2323
Root an. cond. 6.183236.18323
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  − 2.54·5-s − 7-s + 5.35·11-s + 6.80·13-s + 4.03·17-s − 19-s − 7.79·23-s + 1.45·25-s + 2.81·29-s − 1.54·31-s + 2.54·35-s + 5.09·37-s − 9.22·41-s + 1.54·43-s − 5.46·47-s + 49-s + 9.39·53-s − 13.6·55-s + 5.09·61-s − 17.2·65-s − 3.25·67-s + 1.32·71-s + 16.5·73-s − 5.35·77-s + 11.2·79-s − 6.19·83-s − 10.2·85-s + ⋯
L(s)  = 1  − 1.13·5-s − 0.377·7-s + 1.61·11-s + 1.88·13-s + 0.978·17-s − 0.229·19-s − 1.62·23-s + 0.290·25-s + 0.523·29-s − 0.277·31-s + 0.429·35-s + 0.837·37-s − 1.44·41-s + 0.235·43-s − 0.797·47-s + 0.142·49-s + 1.28·53-s − 1.83·55-s + 0.651·61-s − 2.14·65-s − 0.398·67-s + 0.157·71-s + 1.93·73-s − 0.610·77-s + 1.26·79-s − 0.679·83-s − 1.11·85-s + ⋯

Functional equation

Λ(s)=(4788s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4788s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 47884788    =    22⋅32⋅7⋅192^{2} \cdot 3^{2} \cdot 7 \cdot 19
Sign: 11
Analytic conductor: 38.232338.2323
Root analytic conductor: 6.183236.18323
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4788, ( :1/2), 1)(2,\ 4788,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.7310954141.731095414
L(12)L(\frac12) ≈\approx 1.7310954141.731095414
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 1
3 1 1
7 1+T 1 + T
19 1+T 1 + T
good5 1+2.54T+5T2 1 + 2.54T + 5T^{2}
11 1−5.35T+11T2 1 - 5.35T + 11T^{2}
13 1−6.80T+13T2 1 - 6.80T + 13T^{2}
17 1−4.03T+17T2 1 - 4.03T + 17T^{2}
23 1+7.79T+23T2 1 + 7.79T + 23T^{2}
29 1−2.81T+29T2 1 - 2.81T + 29T^{2}
31 1+1.54T+31T2 1 + 1.54T + 31T^{2}
37 1−5.09T+37T2 1 - 5.09T + 37T^{2}
41 1+9.22T+41T2 1 + 9.22T + 41T^{2}
43 1−1.54T+43T2 1 - 1.54T + 43T^{2}
47 1+5.46T+47T2 1 + 5.46T + 47T^{2}
53 1−9.39T+53T2 1 - 9.39T + 53T^{2}
59 1+59T2 1 + 59T^{2}
61 1−5.09T+61T2 1 - 5.09T + 61T^{2}
67 1+3.25T+67T2 1 + 3.25T + 67T^{2}
71 1−1.32T+71T2 1 - 1.32T + 71T^{2}
73 1−16.5T+73T2 1 - 16.5T + 73T^{2}
79 1−11.2T+79T2 1 - 11.2T + 79T^{2}
83 1+6.19T+83T2 1 + 6.19T + 83T^{2}
89 1+11.6T+89T2 1 + 11.6T + 89T^{2}
97 1+17.9T+97T2 1 + 17.9T + 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

−8.299188409379086985163201407174, −7.70978490289662335078219414827, −6.66309533391090313085671419640, −6.31409246288072125601485836429, −5.46708088910072345842167089718, −4.08050975262303829175866257678, −3.91820568119709272520041352285, −3.23757888162928826721489264790, −1.71681658209574510326074413655, −0.76654623292882484452203057510, 0.76654623292882484452203057510, 1.71681658209574510326074413655, 3.23757888162928826721489264790, 3.91820568119709272520041352285, 4.08050975262303829175866257678, 5.46708088910072345842167089718, 6.31409246288072125601485836429, 6.66309533391090313085671419640, 7.70978490289662335078219414827, 8.299188409379086985163201407174

Graph of the ZZ-function along the critical line