↑

Properties

Label 4-2695275-1.1-c1e2-0-48
Degree 44
Conductor 26952752695275
Sign −1-1
Analytic cond. 171.853171.853
Root an. cond. 3.620673.62067
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 2·4-s + 9-s + 11-s + 2·12-s − 10·23-s + 25-s + 27-s − 5·31-s + 33-s + 2·36-s − 2·37-s + 2·44-s − 8·47-s + 49-s + 4·59-s − 8·64-s − 8·67-s − 10·69-s + 18·71-s + 75-s + 81-s − 14·89-s − 20·92-s − 5·93-s + 14·97-s + 99-s + ⋯
L(s)  = 1  + 0.577·3-s + 4-s + 1/3·9-s + 0.301·11-s + 0.577·12-s − 2.08·23-s + 1/5·25-s + 0.192·27-s − 0.898·31-s + 0.174·33-s + 1/3·36-s − 0.328·37-s + 0.301·44-s − 1.16·47-s + 1/7·49-s + 0.520·59-s − 64-s − 0.977·67-s − 1.20·69-s + 2.13·71-s + 0.115·75-s + 1/9·81-s − 1.48·89-s − 2.08·92-s − 0.518·93-s + 1.42·97-s + 0.100·99-s + ⋯

Functional equation

Λ(s)=(2695275s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2695275 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(2695275s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2695275 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 26952752695275    =    34⋅52⋅1133^{4} \cdot 5^{2} \cdot 11^{3}
Sign: −1-1
Analytic conductor: 171.853171.853
Root analytic conductor: 3.620673.62067
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 2695275, ( :1/2,1/2), −1)(4,\ 2695275,\ (\ :1/2, 1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad3C1C_1 1−T 1 - T
5C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
11C1C_1 1−T 1 - T
good2C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
7C22C_2^2 1−T2+p2T4 1 - T^{2} + p^{2} T^{4}
13C22C_2^2 1+11T2+p2T4 1 + 11 T^{2} + p^{2} T^{4}
17C22C_2^2 1+13T2+p2T4 1 + 13 T^{2} + p^{2} T^{4}
19C22C_2^2 1+5T2+p2T4 1 + 5 T^{2} + p^{2} T^{4}
23C2C_2×\timesC2C_2 (1+T+pT2)(1+9T+pT2) ( 1 + T + p T^{2} )( 1 + 9 T + p T^{2} )
29C22C_2^2 1+25T2+p2T4 1 + 25 T^{2} + p^{2} T^{4}
31C2C_2×\timesC2C_2 (1+pT2)(1+5T+pT2) ( 1 + p T^{2} )( 1 + 5 T + p T^{2} )
37C2C_2×\timesC2C_2 (1−6T+pT2)(1+8T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} )
41C22C_2^2 1+74T2+p2T4 1 + 74 T^{2} + p^{2} T^{4}
43C22C_2^2 1−29T2+p2T4 1 - 29 T^{2} + p^{2} T^{4}
47C2C_2×\timesC2C_2 (1+T+pT2)(1+7T+pT2) ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} )
53C2C_2 (1−T+pT2)(1+T+pT2) ( 1 - T + p T^{2} )( 1 + T + p T^{2} )
59C2C_2×\timesC2C_2 (1−14T+pT2)(1+10T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} )
61C22C_2^2 1−65T2+p2T4 1 - 65 T^{2} + p^{2} T^{4}
67C2C_2×\timesC2C_2 (1−2T+pT2)(1+10T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} )
71C2C_2×\timesC2C_2 (1−12T+pT2)(1−6T+pT2) ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} )
73C22C_2^2 1−9T2+p2T4 1 - 9 T^{2} + p^{2} T^{4}
79C22C_2^2 1−15T2+p2T4 1 - 15 T^{2} + p^{2} T^{4}
83C22C_2^2 1+51T2+p2T4 1 + 51 T^{2} + p^{2} T^{4}
89C2C_2×\timesC2C_2 (1+pT2)(1+14T+pT2) ( 1 + p T^{2} )( 1 + 14 T + p T^{2} )
97C2C_2×\timesC2C_2 (1−12T+pT2)(1−2T+pT2) ( 1 - 12 T + p T^{2} )( 1 - 2 T + p T^{2} )
show more
show less
   L(s)=∏p ∏j=14(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−7.30661797541290821455696261649, −7.03189212108301260064437659817, −6.63216653749155622423965381578, −6.23404778308086977014137472094, −5.79202265149903668260362805735, −5.40844333678785302124910491259, −4.70329769108401126404570380961, −4.34563034217913995952762795991, −3.70093089349896061706983027467, −3.45992144781733985221268658583, −2.79639644855330517507098304064, −2.15293534837809612550077905006, −1.96819885237805176422041671183, −1.24271455357801539190571875522, 0, 1.24271455357801539190571875522, 1.96819885237805176422041671183, 2.15293534837809612550077905006, 2.79639644855330517507098304064, 3.45992144781733985221268658583, 3.70093089349896061706983027467, 4.34563034217913995952762795991, 4.70329769108401126404570380961, 5.40844333678785302124910491259, 5.79202265149903668260362805735, 6.23404778308086977014137472094, 6.63216653749155622423965381578, 7.03189212108301260064437659817, 7.30661797541290821455696261649

Graph of the ZZ-function along the critical line