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Properties

Label 4-4416e2-1.1-c1e2-0-4
Degree 44
Conductor 1950105619501056
Sign 11
Analytic cond. 1243.401243.40
Root an. cond. 5.938175.93817
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 2·5-s − 2·7-s + 3·9-s + 8·11-s − 4·15-s − 10·17-s + 10·19-s + 4·21-s − 2·23-s − 2·25-s − 4·27-s + 4·31-s − 16·33-s − 4·35-s − 4·41-s + 2·43-s + 6·45-s + 8·47-s − 6·49-s + 20·51-s + 6·53-s + 16·55-s − 20·57-s + 8·59-s − 6·63-s + 6·67-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.894·5-s − 0.755·7-s + 9-s + 2.41·11-s − 1.03·15-s − 2.42·17-s + 2.29·19-s + 0.872·21-s − 0.417·23-s − 2/5·25-s − 0.769·27-s + 0.718·31-s − 2.78·33-s − 0.676·35-s − 0.624·41-s + 0.304·43-s + 0.894·45-s + 1.16·47-s − 6/7·49-s + 2.80·51-s + 0.824·53-s + 2.15·55-s − 2.64·57-s + 1.04·59-s − 0.755·63-s + 0.733·67-s + ⋯

Functional equation

Λ(s)=(19501056s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 19501056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(19501056s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 19501056 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1950105619501056    =    212⋅32⋅2322^{12} \cdot 3^{2} \cdot 23^{2}
Sign: 11
Analytic conductor: 1243.401243.40
Root analytic conductor: 5.938175.93817
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 19501056, ( :1/2,1/2), 1)(4,\ 19501056,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.6477889082.647788908
L(12)L(\frac12) ≈\approx 2.6477889082.647788908
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
3C1C_1 (1+T)2 ( 1 + T )^{2}
23C1C_1 (1+T)2 ( 1 + T )^{2}
good5D4D_{4} 1−2T+6T2−2pT3+p2T4 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4}
7D4D_{4} 1+2T+10T2+2pT3+p2T4 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4}
11C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
13C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
17D4D_{4} 1+10T+54T2+10pT3+p2T4 1 + 10 T + 54 T^{2} + 10 p T^{3} + p^{2} T^{4}
19D4D_{4} 1−10T+58T2−10pT3+p2T4 1 - 10 T + 58 T^{2} - 10 p T^{3} + p^{2} T^{4}
29C22C_2^2 1+38T2+p2T4 1 + 38 T^{2} + p^{2} T^{4}
31C4C_4 1−4T+46T2−4pT3+p2T4 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4}
37C22C_2^2 1+54T2+p2T4 1 + 54 T^{2} + p^{2} T^{4}
41C4C_4 1+4T+6T2+4pT3+p2T4 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4}
43D4D_{4} 1−2T+42T2−2pT3+p2T4 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4}
47C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
53D4D_{4} 1−6T+110T2−6pT3+p2T4 1 - 6 T + 110 T^{2} - 6 p T^{3} + p^{2} T^{4}
59D4D_{4} 1−8T+54T2−8pT3+p2T4 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4}
61C22C_2^2 1+102T2+p2T4 1 + 102 T^{2} + p^{2} T^{4}
67D4D_{4} 1−6T+138T2−6pT3+p2T4 1 - 6 T + 138 T^{2} - 6 p T^{3} + p^{2} T^{4}
71C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
73D4D_{4} 1+4T+70T2+4pT3+p2T4 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4}
79D4D_{4} 1+6T+122T2+6pT3+p2T4 1 + 6 T + 122 T^{2} + 6 p T^{3} + p^{2} T^{4}
83C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
89D4D_{4} 1−2T+174T2−2pT3+p2T4 1 - 2 T + 174 T^{2} - 2 p T^{3} + p^{2} T^{4}
97D4D_{4} 1−8T+190T2−8pT3+p2T4 1 - 8 T + 190 T^{2} - 8 p T^{3} + p^{2} T^{4}
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   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

−8.507716418999817764574548290266, −8.441408282366711073995019785350, −7.46528762319905767988794167842, −7.40494721912235168355962595950, −6.75262703450261873663320916954, −6.74698706265037305372256343962, −6.28434014368888752179255233395, −6.18206608454870779948807654788, −5.57231049812406729684594275727, −5.45658927915618676565695952104, −4.72794266175950756363362897090, −4.53991319426220476838226849336, −4.06345242718058504815478985112, −3.58506089018139398996366427890, −3.36601026636722088450397073921, −2.55101066063760646660556144537, −1.98720633005334702452871899136, −1.73023988982698844854951473365, −0.898696741706639839845683108283, −0.63678739587641530320474763007, 0.63678739587641530320474763007, 0.898696741706639839845683108283, 1.73023988982698844854951473365, 1.98720633005334702452871899136, 2.55101066063760646660556144537, 3.36601026636722088450397073921, 3.58506089018139398996366427890, 4.06345242718058504815478985112, 4.53991319426220476838226849336, 4.72794266175950756363362897090, 5.45658927915618676565695952104, 5.57231049812406729684594275727, 6.18206608454870779948807654788, 6.28434014368888752179255233395, 6.74698706265037305372256343962, 6.75262703450261873663320916954, 7.40494721912235168355962595950, 7.46528762319905767988794167842, 8.441408282366711073995019785350, 8.507716418999817764574548290266

Graph of the ZZ-function along the critical line