↑

Properties

Label 2-1350-1.1-c1-0-13
Degree 22
Conductor 13501350
Sign 11
Analytic cond. 10.779810.7798
Root an. cond. 3.283263.28326
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 4.35·7-s + 8-s − 4.35·11-s + 4.35·14-s + 16-s + 4·17-s + 6·19-s − 4.35·22-s − 2·23-s + 4.35·28-s + 7·31-s + 32-s + 4·34-s − 8.71·37-s + 6·38-s + 8.71·41-s − 8.71·43-s − 4.35·44-s − 2·46-s + 2·47-s + 12.0·49-s − 3·53-s + 4.35·56-s − 8.71·59-s − 4·61-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 1.64·7-s + 0.353·8-s − 1.31·11-s + 1.16·14-s + 0.250·16-s + 0.970·17-s + 1.37·19-s − 0.929·22-s − 0.417·23-s + 0.823·28-s + 1.25·31-s + 0.176·32-s + 0.685·34-s − 1.43·37-s + 0.973·38-s + 1.36·41-s − 1.32·43-s − 0.657·44-s − 0.294·46-s + 0.291·47-s + 1.71·49-s − 0.412·53-s + 0.582·56-s − 1.13·59-s − 0.512·61-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: no
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 3.0754186663.075418666
L(12)L(\frac12) ≈\approx 3.0754186663.075418666
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 1−4.35T+7T2 1 - 4.35T + 7T^{2}
11 1+4.35T+11T2 1 + 4.35T + 11T^{2}
13 1+13T2 1 + 13T^{2}
17 1−4T+17T2 1 - 4T + 17T^{2}
19 1−6T+19T2 1 - 6T + 19T^{2}
23 1+2T+23T2 1 + 2T + 23T^{2}
29 1+29T2 1 + 29T^{2}
31 1−7T+31T2 1 - 7T + 31T^{2}
37 1+8.71T+37T2 1 + 8.71T + 37T^{2}
41 1−8.71T+41T2 1 - 8.71T + 41T^{2}
43 1+8.71T+43T2 1 + 8.71T + 43T^{2}
47 1−2T+47T2 1 - 2T + 47T^{2}
53 1+3T+53T2 1 + 3T + 53T^{2}
59 1+8.71T+59T2 1 + 8.71T + 59T^{2}
61 1+4T+61T2 1 + 4T + 61T^{2}
67 1−8.71T+67T2 1 - 8.71T + 67T^{2}
71 1+71T2 1 + 71T^{2}
73 1−4.35T+73T2 1 - 4.35T + 73T^{2}
79 1+79T2 1 + 79T^{2}
83 1+5T+83T2 1 + 5T + 83T^{2}
89 1−8.71T+89T2 1 - 8.71T + 89T^{2}
97 1+4.35T+97T2 1 + 4.35T + 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

−9.831766250536615909555991793629, −8.521640902459070146262123743588, −7.80286480238678608669437815722, −7.41152490360258934930589333356, −6.03490131305999332310631451392, −5.12800961747859868254216974970, −4.85036360342555564090829170734, −3.53307271168479685023107989173, −2.48726936281705293448809254923, −1.31472464666321829271225485775, 1.31472464666321829271225485775, 2.48726936281705293448809254923, 3.53307271168479685023107989173, 4.85036360342555564090829170734, 5.12800961747859868254216974970, 6.03490131305999332310631451392, 7.41152490360258934930589333356, 7.80286480238678608669437815722, 8.521640902459070146262123743588, 9.831766250536615909555991793629

Graph of the ZZ-function along the critical line