↑

Properties

Label 2-435-435.434-c2-0-59
Degree 22
Conductor 435435
Sign 11
Analytic cond. 11.852811.8528
Root an. cond. 3.442803.44280
Motivic weight 22
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 4·4-s + 5·5-s + 9·9-s + 7·11-s − 12·12-s − 15·15-s + 16·16-s + 20·20-s − 41·23-s + 25·25-s − 27·27-s + 29·29-s − 21·33-s + 36·36-s + 71·37-s − 53·41-s + 59·43-s + 28·44-s + 45·45-s − 48·48-s + 49·49-s + 19·53-s + 35·55-s − 60·60-s + 64·64-s + 123·69-s + ⋯
L(s)  = 1  − 3-s + 4-s + 5-s + 9-s + 7/11·11-s − 12-s − 15-s + 16-s + 20-s − 1.78·23-s + 25-s − 27-s + 29-s − 0.636·33-s + 36-s + 1.91·37-s − 1.29·41-s + 1.37·43-s + 7/11·44-s + 45-s − 48-s + 49-s + 0.358·53-s + 7/11·55-s − 60-s + 64-s + 1.78·69-s + ⋯

Functional equation

Λ(s)=(435s/2 ΓC(s) L(s)=( Λ(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}
Λ(s)=(435s/2 ΓC(s+1) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 435435    =    3⋅5⋅293 \cdot 5 \cdot 29
Sign: 11
Analytic conductor: 11.852811.8528
Root analytic conductor: 3.442803.44280
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: χ435(434,⋅)\chi_{435} (434, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 435, ( :1), 1)(2,\ 435,\ (\ :1),\ 1)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 2.0120133712.012013371
L(12)L(\frac12) ≈\approx 2.0120133712.012013371
L(2)L(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+pT 1 + p T
5 1−pT 1 - p T
29 1−pT 1 - p T
good2 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
7 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
11 1−7T+p2T2 1 - 7 T + p^{2} T^{2}
13 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
17 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
19 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
23 1+41T+p2T2 1 + 41 T + p^{2} T^{2}
31 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
37 1−71T+p2T2 1 - 71 T + p^{2} T^{2}
41 1+53T+p2T2 1 + 53 T + p^{2} T^{2}
43 1−59T+p2T2 1 - 59 T + p^{2} T^{2}
47 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
53 1−19T+p2T2 1 - 19 T + p^{2} T^{2}
59 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
61 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
67 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
71 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
73 1+T+p2T2 1 + T + p^{2} T^{2}
79 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
83 1−79T+p2T2 1 - 79 T + p^{2} T^{2}
89 1+62T+p2T2 1 + 62 T + p^{2} T^{2}
97 1+49T+p2T2 1 + 49 T + p^{2} 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

−10.88965458956183607328413247932, −10.20844562524526650929498218394, −9.468201675371588189844156313502, −8.002005883923759507259649282931, −6.84859645840711605667824252059, −6.21015608819662411179756023276, −5.53425175704503221255828643468, −4.14738178195932642111416106758, −2.42269363291139208172532716134, −1.23528383259415825245829677177, 1.23528383259415825245829677177, 2.42269363291139208172532716134, 4.14738178195932642111416106758, 5.53425175704503221255828643468, 6.21015608819662411179756023276, 6.84859645840711605667824252059, 8.002005883923759507259649282931, 9.468201675371588189844156313502, 10.20844562524526650929498218394, 10.88965458956183607328413247932

Graph of the ZZ-function along the critical line