↑

Properties

Label 2-1350-1.1-c1-0-8
Degree 22
Conductor 13501350
Sign 11
Analytic cond. 10.779810.7798
Root an. cond. 3.283263.28326
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 2·7-s + 8-s + 3·11-s + 13-s − 2·14-s + 16-s − 3·17-s + 8·19-s + 3·22-s + 3·23-s + 26-s − 2·28-s + 9·29-s − 7·31-s + 32-s − 3·34-s − 2·37-s + 8·38-s + 12·41-s + 7·43-s + 3·44-s + 3·46-s − 3·47-s − 3·49-s + 52-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.755·7-s + 0.353·8-s + 0.904·11-s + 0.277·13-s − 0.534·14-s + 1/4·16-s − 0.727·17-s + 1.83·19-s + 0.639·22-s + 0.625·23-s + 0.196·26-s − 0.377·28-s + 1.67·29-s − 1.25·31-s + 0.176·32-s − 0.514·34-s − 0.328·37-s + 1.29·38-s + 1.87·41-s + 1.06·43-s + 0.452·44-s + 0.442·46-s − 0.437·47-s − 3/7·49-s + 0.138·52-s + ⋯

Functional equation

Λ(s)=(1350s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1350s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13501350    =    2⋅33⋅522 \cdot 3^{3} \cdot 5^{2}
Sign: 11
Analytic conductor: 10.779810.7798
Root analytic conductor: 3.283263.28326
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1350, ( :1/2), 1)(2,\ 1350,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.6487855632.648785563
L(12)L(\frac12) ≈\approx 2.6487855632.648785563
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
good7 1+2T+pT2 1 + 2 T + p T^{2}
11 1−3T+pT2 1 - 3 T + p T^{2}
13 1−T+pT2 1 - T + p T^{2}
17 1+3T+pT2 1 + 3 T + p T^{2}
19 1−8T+pT2 1 - 8 T + p T^{2}
23 1−3T+pT2 1 - 3 T + p T^{2}
29 1−9T+pT2 1 - 9 T + p T^{2}
31 1+7T+pT2 1 + 7 T + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1−12T+pT2 1 - 12 T + p T^{2}
43 1−7T+pT2 1 - 7 T + p T^{2}
47 1+3T+pT2 1 + 3 T + p T^{2}
53 1−12T+pT2 1 - 12 T + p T^{2}
59 1+12T+pT2 1 + 12 T + p T^{2}
61 1+10T+pT2 1 + 10 T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1+2T+pT2 1 + 2 T + p T^{2}
79 1+T+pT2 1 + T + p T^{2}
83 1−18T+pT2 1 - 18 T + p T^{2}
89 1+pT2 1 + p T^{2}
97 1+14T+pT2 1 + 14 T + p T^{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

−9.429245595815721672181577127890, −9.072503855710526131661237974419, −7.77525488848171336433338902364, −6.96206917457341163684760702823, −6.29697405356291031337465683385, −5.44624806543657551278179807260, −4.42905986906434781866538981809, −3.52758168075608665434978537597, −2.69679316478050875122526015205, −1.15765950034440094850851077493, 1.15765950034440094850851077493, 2.69679316478050875122526015205, 3.52758168075608665434978537597, 4.42905986906434781866538981809, 5.44624806543657551278179807260, 6.29697405356291031337465683385, 6.96206917457341163684760702823, 7.77525488848171336433338902364, 9.072503855710526131661237974419, 9.429245595815721672181577127890

Graph of the ZZ-function along the critical line