↑

Properties

Label 2-315-35.34-c2-0-15
Degree 22
Conductor 315315
Sign 11
Analytic cond. 8.583128.58312
Root an. cond. 2.929692.92969
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  + 4·4-s − 5·5-s − 7·7-s + 13·11-s + 19·13-s + 16·16-s + 29·17-s − 20·20-s + 25·25-s − 28·28-s − 23·29-s + 35·35-s + 52·44-s − 31·47-s + 49·49-s + 76·52-s − 65·55-s + 64·64-s − 95·65-s + 116·68-s − 2·71-s + 34·73-s − 91·77-s − 157·79-s − 80·80-s + 86·83-s − 145·85-s + ⋯
L(s)  = 1  + 4-s − 5-s − 7-s + 1.18·11-s + 1.46·13-s + 16-s + 1.70·17-s − 20-s + 25-s − 28-s − 0.793·29-s + 35-s + 1.18·44-s − 0.659·47-s + 49-s + 1.46·52-s − 1.18·55-s + 64-s − 1.46·65-s + 1.70·68-s − 0.0281·71-s + 0.465·73-s − 1.18·77-s − 1.98·79-s − 80-s + 1.03·83-s − 1.70·85-s + ⋯

Functional equation

Λ(s)=(315s/2 ΓC(s) L(s)=( Λ(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}
Λ(s)=(315s/2 ΓC(s+1) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 315315    =    32⋅5⋅73^{2} \cdot 5 \cdot 7
Sign: 11
Analytic conductor: 8.583128.58312
Root analytic conductor: 2.929692.92969
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: χ315(244,⋅)\chi_{315} (244, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 315, ( :1), 1)(2,\ 315,\ (\ :1),\ 1)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.8139164911.813916491
L(12)L(\frac12) ≈\approx 1.8139164911.813916491
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 1
5 1+pT 1 + p T
7 1+pT 1 + p T
good2 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
11 1−13T+p2T2 1 - 13 T + p^{2} T^{2}
13 1−19T+p2T2 1 - 19 T + p^{2} T^{2}
17 1−29T+p2T2 1 - 29 T + p^{2} T^{2}
19 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
23 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
29 1+23T+p2T2 1 + 23 T + p^{2} T^{2}
31 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
37 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
41 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
43 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
47 1+31T+p2T2 1 + 31 T + p^{2} T^{2}
53 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
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+2T+p2T2 1 + 2 T + p^{2} T^{2}
73 1−34T+p2T2 1 - 34 T + p^{2} T^{2}
79 1+157T+p2T2 1 + 157 T + p^{2} T^{2}
83 1−86T+p2T2 1 - 86 T + p^{2} T^{2}
89 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
97 1+149T+p2T2 1 + 149 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

−11.53909938078069870630499054908, −10.69010714352588202670322378227, −9.636233618004979992010515391987, −8.500550253919827948579538127354, −7.48630701906346438116537150418, −6.60451494626098115725564219985, −5.78598926920184600444775710633, −3.81242820160127633922274281554, −3.24380662599066294345935222613, −1.19229059463459525297986245435, 1.19229059463459525297986245435, 3.24380662599066294345935222613, 3.81242820160127633922274281554, 5.78598926920184600444775710633, 6.60451494626098115725564219985, 7.48630701906346438116537150418, 8.500550253919827948579538127354, 9.636233618004979992010515391987, 10.69010714352588202670322378227, 11.53909938078069870630499054908

Graph of the ZZ-function along the critical line