↑

Properties

Label 4-936e2-1.1-c1e2-0-1
Degree 44
Conductor 876096876096
Sign 11
Analytic cond. 55.860655.8606
Root an. cond. 2.733862.73386
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 6·5-s + 4·7-s + 4·11-s − 5·13-s + 3·17-s + 4·19-s − 8·23-s + 17·25-s − 5·29-s − 16·31-s − 24·35-s − 7·37-s − 9·41-s − 8·43-s + 8·47-s + 7·49-s + 10·53-s − 24·55-s + 4·59-s + 5·61-s + 30·65-s − 8·67-s − 4·71-s + 22·73-s + 16·77-s − 8·79-s − 18·85-s + ⋯
L(s)  = 1  − 2.68·5-s + 1.51·7-s + 1.20·11-s − 1.38·13-s + 0.727·17-s + 0.917·19-s − 1.66·23-s + 17/5·25-s − 0.928·29-s − 2.87·31-s − 4.05·35-s − 1.15·37-s − 1.40·41-s − 1.21·43-s + 1.16·47-s + 49-s + 1.37·53-s − 3.23·55-s + 0.520·59-s + 0.640·61-s + 3.72·65-s − 0.977·67-s − 0.474·71-s + 2.57·73-s + 1.82·77-s − 0.900·79-s − 1.95·85-s + ⋯

Functional equation

Λ(s)=(876096s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 876096 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(876096s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 876096 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 876096876096    =    26⋅34⋅1322^{6} \cdot 3^{4} \cdot 13^{2}
Sign: 11
Analytic conductor: 55.860655.8606
Root analytic conductor: 2.733862.73386
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 876096, ( :1/2,1/2), 1)(4,\ 876096,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.69288716980.6928871698
L(12)L(\frac12) ≈\approx 0.69288716980.6928871698
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)
bad2 1 1
3 1 1
13C2C_2 1+5T+pT2 1 + 5 T + p T^{2}
good5C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
7C2C_2 (1−5T+pT2)(1+T+pT2) ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} )
11C22C_2^2 1−4T+5T2−4pT3+p2T4 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4}
17C22C_2^2 1−3T−8T2−3pT3+p2T4 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4}
19C22C_2^2 1−4T−3T2−4pT3+p2T4 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4}
23C22C_2^2 1+8T+41T2+8pT3+p2T4 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4}
29C22C_2^2 1+5T−4T2+5pT3+p2T4 1 + 5 T - 4 T^{2} + 5 p T^{3} + p^{2} T^{4}
31C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
37C22C_2^2 1+7T+12T2+7pT3+p2T4 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4}
41C22C_2^2 1+9T+40T2+9pT3+p2T4 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4}
43C2C_2 (1−5T+pT2)(1+13T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} )
47C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
53C2C_2 (1−5T+pT2)2 ( 1 - 5 T + p T^{2} )^{2}
59C22C_2^2 1−4T−43T2−4pT3+p2T4 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4}
61C22C_2^2 1−5T−36T2−5pT3+p2T4 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4}
67C22C_2^2 1+8T−3T2+8pT3+p2T4 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4}
71C22C_2^2 1+4T−55T2+4pT3+p2T4 1 + 4 T - 55 T^{2} + 4 p T^{3} + p^{2} T^{4}
73C2C_2 (1−11T+pT2)2 ( 1 - 11 T + p T^{2} )^{2}
79C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
83C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
89C22C_2^2 1+6T−53T2+6pT3+p2T4 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4}
97C2C_2 (1−19T+pT2)(1+5T+pT2) ( 1 - 19 T + p T^{2} )( 1 + 5 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

−10.44576908600006344510224581554, −9.831099065397905593927316513367, −9.339559828881023943090777047827, −8.926709517664852580663735112682, −8.408689480637070077514118951523, −8.045785227277115288864906260437, −7.81178514871832325268213458267, −7.32847919310981351068343944314, −7.16333818514872748276182623727, −6.80406900819133450422673111009, −5.72284034130761734054914096761, −5.31308900094643306220052118701, −5.03438953992755654583183824456, −4.30107391680632332291633076927, −3.86837836122392383913578565062, −3.76094796415234917020729315861, −3.21063136376065098437512599043, −2.03891394506725681742493250815, −1.60646807934623313479652226127, −0.39906640497337272797831426704, 0.39906640497337272797831426704, 1.60646807934623313479652226127, 2.03891394506725681742493250815, 3.21063136376065098437512599043, 3.76094796415234917020729315861, 3.86837836122392383913578565062, 4.30107391680632332291633076927, 5.03438953992755654583183824456, 5.31308900094643306220052118701, 5.72284034130761734054914096761, 6.80406900819133450422673111009, 7.16333818514872748276182623727, 7.32847919310981351068343944314, 7.81178514871832325268213458267, 8.045785227277115288864906260437, 8.408689480637070077514118951523, 8.926709517664852580663735112682, 9.339559828881023943090777047827, 9.831099065397905593927316513367, 10.44576908600006344510224581554

Graph of the ZZ-function along the critical line