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Properties

Label 2-4788-1.1-c1-0-21
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  + 4.21·5-s + 7-s − 2·11-s + 0.643·13-s + 4.21·17-s − 19-s − 2·23-s + 12.7·25-s − 4.93·29-s + 1.35·31-s + 4.21·35-s + 10.4·37-s + 4·41-s − 5.79·43-s − 5.50·47-s + 49-s − 6.21·53-s − 8.43·55-s + 14.4·61-s + 2.71·65-s + 8.43·67-s + 6.21·71-s + 3.28·73-s − 2·77-s + 15.1·79-s − 2.93·83-s + 17.7·85-s + ⋯
L(s)  = 1  + 1.88·5-s + 0.377·7-s − 0.603·11-s + 0.178·13-s + 1.02·17-s − 0.229·19-s − 0.417·23-s + 2.55·25-s − 0.915·29-s + 0.243·31-s + 0.713·35-s + 1.71·37-s + 0.624·41-s − 0.883·43-s − 0.802·47-s + 0.142·49-s − 0.854·53-s − 1.13·55-s + 1.84·61-s + 0.336·65-s + 1.03·67-s + 0.737·71-s + 0.384·73-s − 0.227·77-s + 1.70·79-s − 0.321·83-s + 1.92·85-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 3.1235706343.123570634
L(12)L(\frac12) ≈\approx 3.1235706343.123570634
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 1−4.21T+5T2 1 - 4.21T + 5T^{2}
11 1+2T+11T2 1 + 2T + 11T^{2}
13 1−0.643T+13T2 1 - 0.643T + 13T^{2}
17 1−4.21T+17T2 1 - 4.21T + 17T^{2}
23 1+2T+23T2 1 + 2T + 23T^{2}
29 1+4.93T+29T2 1 + 4.93T + 29T^{2}
31 1−1.35T+31T2 1 - 1.35T + 31T^{2}
37 1−10.4T+37T2 1 - 10.4T + 37T^{2}
41 1−4T+41T2 1 - 4T + 41T^{2}
43 1+5.79T+43T2 1 + 5.79T + 43T^{2}
47 1+5.50T+47T2 1 + 5.50T + 47T^{2}
53 1+6.21T+53T2 1 + 6.21T + 53T^{2}
59 1+59T2 1 + 59T^{2}
61 1−14.4T+61T2 1 - 14.4T + 61T^{2}
67 1−8.43T+67T2 1 - 8.43T + 67T^{2}
71 1−6.21T+71T2 1 - 6.21T + 71T^{2}
73 1−3.28T+73T2 1 - 3.28T + 73T^{2}
79 1−15.1T+79T2 1 - 15.1T + 79T^{2}
83 1+2.93T+83T2 1 + 2.93T + 83T^{2}
89 1−4T+89T2 1 - 4T + 89T^{2}
97 1+2T+97T2 1 + 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

−8.246812586254922099049160179159, −7.67029842533276008740202102884, −6.60591933293205581966952345133, −6.07163969672511990257431250265, −5.37356461999106883584944244057, −4.90073649282574003463116397172, −3.67499095781995070956076402195, −2.61492271612151799254123814915, −1.98789198409049578848396519701, −1.03180243000369693299706856890, 1.03180243000369693299706856890, 1.98789198409049578848396519701, 2.61492271612151799254123814915, 3.67499095781995070956076402195, 4.90073649282574003463116397172, 5.37356461999106883584944244057, 6.07163969672511990257431250265, 6.60591933293205581966952345133, 7.67029842533276008740202102884, 8.246812586254922099049160179159

Graph of the ZZ-function along the critical line