↑

Properties

Label 2-1323-1.1-c3-0-24
Degree 22
Conductor 13231323
Sign 11
Analytic cond. 78.059578.0595
Root an. cond. 8.835138.83513
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.959·2-s − 7.07·4-s − 10.2·5-s + 14.4·8-s + 9.87·10-s + 28.3·11-s − 5.96·13-s + 42.7·16-s + 104.·17-s − 34.0·19-s + 72.8·20-s − 27.1·22-s − 103.·23-s − 19.1·25-s + 5.72·26-s − 195.·29-s − 9.17·31-s − 156.·32-s − 100.·34-s + 245.·37-s + 32.6·38-s − 148.·40-s − 366.·41-s − 366.·43-s − 200.·44-s + 99.4·46-s − 244.·47-s + ⋯
L(s)  = 1  − 0.339·2-s − 0.884·4-s − 0.920·5-s + 0.639·8-s + 0.312·10-s + 0.776·11-s − 0.127·13-s + 0.667·16-s + 1.49·17-s − 0.411·19-s + 0.814·20-s − 0.263·22-s − 0.939·23-s − 0.152·25-s + 0.0431·26-s − 1.25·29-s − 0.0531·31-s − 0.866·32-s − 0.506·34-s + 1.09·37-s + 0.139·38-s − 0.588·40-s − 1.39·41-s − 1.29·43-s − 0.687·44-s + 0.318·46-s − 0.758·47-s + ⋯

Functional equation

Λ(s)=(1323s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1323s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13231323    =    33⋅723^{3} \cdot 7^{2}
Sign: 11
Analytic conductor: 78.059578.0595
Root analytic conductor: 8.835138.83513
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1323, ( :3/2), 1)(2,\ 1323,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 0.83117843890.8311784389
L(12)L(\frac12) ≈\approx 0.83117843890.8311784389
L(52)L(\frac{5}{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)
bad3 1 1
7 1 1
good2 1+0.959T+8T2 1 + 0.959T + 8T^{2}
5 1+10.2T+125T2 1 + 10.2T + 125T^{2}
11 1−28.3T+1.33e3T2 1 - 28.3T + 1.33e3T^{2}
13 1+5.96T+2.19e3T2 1 + 5.96T + 2.19e3T^{2}
17 1−104.T+4.91e3T2 1 - 104.T + 4.91e3T^{2}
19 1+34.0T+6.85e3T2 1 + 34.0T + 6.85e3T^{2}
23 1+103.T+1.21e4T2 1 + 103.T + 1.21e4T^{2}
29 1+195.T+2.43e4T2 1 + 195.T + 2.43e4T^{2}
31 1+9.17T+2.97e4T2 1 + 9.17T + 2.97e4T^{2}
37 1−245.T+5.06e4T2 1 - 245.T + 5.06e4T^{2}
41 1+366.T+6.89e4T2 1 + 366.T + 6.89e4T^{2}
43 1+366.T+7.95e4T2 1 + 366.T + 7.95e4T^{2}
47 1+244.T+1.03e5T2 1 + 244.T + 1.03e5T^{2}
53 1−281.T+1.48e5T2 1 - 281.T + 1.48e5T^{2}
59 1−181.T+2.05e5T2 1 - 181.T + 2.05e5T^{2}
61 1−24.1T+2.26e5T2 1 - 24.1T + 2.26e5T^{2}
67 1+336.T+3.00e5T2 1 + 336.T + 3.00e5T^{2}
71 1−196.T+3.57e5T2 1 - 196.T + 3.57e5T^{2}
73 1−683.T+3.89e5T2 1 - 683.T + 3.89e5T^{2}
79 1−619.T+4.93e5T2 1 - 619.T + 4.93e5T^{2}
83 1−176.T+5.71e5T2 1 - 176.T + 5.71e5T^{2}
89 1−761.T+7.04e5T2 1 - 761.T + 7.04e5T^{2}
97 1−1.27e3T+9.12e5T2 1 - 1.27e3T + 9.12e5T^{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

−9.352057427645012862769541472919, −8.228422299111423589581516694042, −7.978191715923465048146161206937, −7.01125628679825607009602591911, −5.86782068841014698626448530484, −4.94343684859852725027966272725, −3.94063277071945029910019992598, −3.48045965269665244402197762674, −1.69223889413001091209951861786, −0.49965696433510390071892428664, 0.49965696433510390071892428664, 1.69223889413001091209951861786, 3.48045965269665244402197762674, 3.94063277071945029910019992598, 4.94343684859852725027966272725, 5.86782068841014698626448530484, 7.01125628679825607009602591911, 7.978191715923465048146161206937, 8.228422299111423589581516694042, 9.352057427645012862769541472919

Graph of the ZZ-function along the critical line