↑

Properties

Label 2-352-1.1-c1-0-4
Degree 22
Conductor 352352
Sign 11
Analytic cond. 2.810732.81073
Root an. cond. 1.676521.67652
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  + 1.56·3-s + 3.56·5-s − 0.561·9-s − 11-s + 2·13-s + 5.56·15-s − 1.12·17-s − 7.12·19-s − 4.68·23-s + 7.68·25-s − 5.56·27-s − 1.12·29-s + 9.56·31-s − 1.56·33-s + 6.68·37-s + 3.12·39-s + 8.24·41-s − 7.12·43-s − 2·45-s + 4·47-s − 7·49-s − 1.75·51-s − 8.24·53-s − 3.56·55-s − 11.1·57-s + 12.6·59-s − 15.3·61-s + ⋯
L(s)  = 1  + 0.901·3-s + 1.59·5-s − 0.187·9-s − 0.301·11-s + 0.554·13-s + 1.43·15-s − 0.272·17-s − 1.63·19-s − 0.976·23-s + 1.53·25-s − 1.07·27-s − 0.208·29-s + 1.71·31-s − 0.271·33-s + 1.09·37-s + 0.500·39-s + 1.28·41-s − 1.08·43-s − 0.298·45-s + 0.583·47-s − 49-s − 0.245·51-s − 1.13·53-s − 0.480·55-s − 1.47·57-s + 1.65·59-s − 1.96·61-s + ⋯

Functional equation

Λ(s)=(352s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(352s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 352352    =    25⋅112^{5} \cdot 11
Sign: 11
Analytic conductor: 2.810732.81073
Root analytic conductor: 1.676521.67652
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 352, ( :1/2), 1)(2,\ 352,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.0753250872.075325087
L(12)L(\frac12) ≈\approx 2.0753250872.075325087
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
11 1+T 1 + T
good3 1−1.56T+3T2 1 - 1.56T + 3T^{2}
5 1−3.56T+5T2 1 - 3.56T + 5T^{2}
7 1+7T2 1 + 7T^{2}
13 1−2T+13T2 1 - 2T + 13T^{2}
17 1+1.12T+17T2 1 + 1.12T + 17T^{2}
19 1+7.12T+19T2 1 + 7.12T + 19T^{2}
23 1+4.68T+23T2 1 + 4.68T + 23T^{2}
29 1+1.12T+29T2 1 + 1.12T + 29T^{2}
31 1−9.56T+31T2 1 - 9.56T + 31T^{2}
37 1−6.68T+37T2 1 - 6.68T + 37T^{2}
41 1−8.24T+41T2 1 - 8.24T + 41T^{2}
43 1+7.12T+43T2 1 + 7.12T + 43T^{2}
47 1−4T+47T2 1 - 4T + 47T^{2}
53 1+8.24T+53T2 1 + 8.24T + 53T^{2}
59 1−12.6T+59T2 1 - 12.6T + 59T^{2}
61 1+15.3T+61T2 1 + 15.3T + 61T^{2}
67 1−4.68T+67T2 1 - 4.68T + 67T^{2}
71 1+3.31T+71T2 1 + 3.31T + 71T^{2}
73 1+6T+73T2 1 + 6T + 73T^{2}
79 1−4.87T+79T2 1 - 4.87T + 79T^{2}
83 1+13.3T+83T2 1 + 13.3T + 83T^{2}
89 1+3.56T+89T2 1 + 3.56T + 89T^{2}
97 1+6.68T+97T2 1 + 6.68T + 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

−11.32072987439750756990133324195, −10.31974112410654281292072961653, −9.600970605761722555739124403564, −8.720317118500883302013183754184, −8.017846064032105851810084781806, −6.42574959118232184729111613601, −5.85574000121122320487660419198, −4.39438615218903394977724440261, −2.81883750856868059259196740156, −1.93278793289298093103504980717, 1.93278793289298093103504980717, 2.81883750856868059259196740156, 4.39438615218903394977724440261, 5.85574000121122320487660419198, 6.42574959118232184729111613601, 8.017846064032105851810084781806, 8.720317118500883302013183754184, 9.600970605761722555739124403564, 10.31974112410654281292072961653, 11.32072987439750756990133324195

Graph of the ZZ-function along the critical line