↑

Properties

Label 2-4788-1.1-c1-0-24
Degree 22
Conductor 47884788
Sign −1-1
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 11

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3.43·5-s − 7-s − 2.61·11-s + 5.43·13-s + 0.611·17-s − 19-s + 1.43·23-s + 6.79·25-s − 1.74·29-s + 0.255·31-s + 3.43·35-s + 5.79·37-s − 8.96·41-s + 7.58·43-s + 10.6·47-s + 49-s − 5.53·53-s + 8.96·55-s − 4.30·59-s − 1.27·61-s − 18.6·65-s + 0.611·67-s + 1.07·71-s − 11.6·73-s + 2.61·77-s − 12.4·79-s + 10.4·83-s + ⋯
L(s)  = 1  − 1.53·5-s − 0.377·7-s − 0.787·11-s + 1.50·13-s + 0.148·17-s − 0.229·19-s + 0.298·23-s + 1.35·25-s − 0.323·29-s + 0.0458·31-s + 0.580·35-s + 0.951·37-s − 1.40·41-s + 1.15·43-s + 1.55·47-s + 0.142·49-s − 0.760·53-s + 1.20·55-s − 0.559·59-s − 0.163·61-s − 2.31·65-s + 0.0747·67-s + 0.127·71-s − 1.36·73-s + 0.297·77-s − 1.40·79-s + 1.14·83-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: −1-1
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: 11
Selberg data: (2, 4788, ( :1/2), −1)(2,\ 4788,\ (\ :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 1
3 1 1
7 1+T 1 + T
19 1+T 1 + T
good5 1+3.43T+5T2 1 + 3.43T + 5T^{2}
11 1+2.61T+11T2 1 + 2.61T + 11T^{2}
13 1−5.43T+13T2 1 - 5.43T + 13T^{2}
17 1−0.611T+17T2 1 - 0.611T + 17T^{2}
23 1−1.43T+23T2 1 - 1.43T + 23T^{2}
29 1+1.74T+29T2 1 + 1.74T + 29T^{2}
31 1−0.255T+31T2 1 - 0.255T + 31T^{2}
37 1−5.79T+37T2 1 - 5.79T + 37T^{2}
41 1+8.96T+41T2 1 + 8.96T + 41T^{2}
43 1−7.58T+43T2 1 - 7.58T + 43T^{2}
47 1−10.6T+47T2 1 - 10.6T + 47T^{2}
53 1+5.53T+53T2 1 + 5.53T + 53T^{2}
59 1+4.30T+59T2 1 + 4.30T + 59T^{2}
61 1+1.27T+61T2 1 + 1.27T + 61T^{2}
67 1−0.611T+67T2 1 - 0.611T + 67T^{2}
71 1−1.07T+71T2 1 - 1.07T + 71T^{2}
73 1+11.6T+73T2 1 + 11.6T + 73T^{2}
79 1+12.4T+79T2 1 + 12.4T + 79T^{2}
83 1−10.4T+83T2 1 - 10.4T + 83T^{2}
89 1−13.2T+89T2 1 - 13.2T + 89T^{2}
97 1+7.88T+97T2 1 + 7.88T + 97T^{2}
show more
show less
   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

−7.80233357009246873797216254831, −7.45709637976400741588544457565, −6.48949630681350100625244847123, −5.81180814346290804813736336848, −4.83133918485081641807896272980, −4.01055706408249825884400235926, −3.48509581821501164855341846140, −2.63091160861477335470633531621, −1.14207886106608566229236838372, 0, 1.14207886106608566229236838372, 2.63091160861477335470633531621, 3.48509581821501164855341846140, 4.01055706408249825884400235926, 4.83133918485081641807896272980, 5.81180814346290804813736336848, 6.48949630681350100625244847123, 7.45709637976400741588544457565, 7.80233357009246873797216254831

Graph of the ZZ-function along the critical line