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Properties

Label 4-3744e2-1.1-c1e2-0-3
Degree 44
Conductor 1401753614017536
Sign 11
Analytic cond. 893.770893.770
Root an. cond. 5.467725.46772
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  − 6·5-s − 2·13-s + 6·17-s + 17·25-s − 20·29-s + 6·37-s − 9·49-s − 8·53-s + 12·65-s + 28·73-s − 36·85-s + 20·89-s − 4·97-s + 24·101-s + 22·109-s + 28·113-s − 2·121-s − 18·125-s + 127-s + 131-s + 137-s + 139-s + 120·145-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  − 2.68·5-s − 0.554·13-s + 1.45·17-s + 17/5·25-s − 3.71·29-s + 0.986·37-s − 9/7·49-s − 1.09·53-s + 1.48·65-s + 3.27·73-s − 3.90·85-s + 2.11·89-s − 0.406·97-s + 2.38·101-s + 2.10·109-s + 2.63·113-s − 0.181·121-s − 1.60·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 9.96·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯

Functional equation

Λ(s)=(14017536s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 14017536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(14017536s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 14017536 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1401753614017536    =    210⋅34⋅1322^{10} \cdot 3^{4} \cdot 13^{2}
Sign: 11
Analytic conductor: 893.770893.770
Root analytic conductor: 5.467725.46772
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 14017536, ( :1/2,1/2), 1)(4,\ 14017536,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.88335641260.8833564126
L(12)L(\frac12) ≈\approx 0.88335641260.8833564126
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
13C1C_1 (1+T)2 ( 1 + T )^{2}
good5C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
7C22C_2^2 1+9T2+p2T4 1 + 9 T^{2} + p^{2} T^{4}
11C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
17C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
19C22C_2^2 1+18T2+p2T4 1 + 18 T^{2} + p^{2} T^{4}
23C22C_2^2 1−34T2+p2T4 1 - 34 T^{2} + p^{2} T^{4}
29C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
31C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
37C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
41C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
43C22C_2^2 1+41T2+p2T4 1 + 41 T^{2} + p^{2} T^{4}
47C22C_2^2 1+89T2+p2T4 1 + 89 T^{2} + p^{2} T^{4}
53C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
59C22C_2^2 1+98T2+p2T4 1 + 98 T^{2} + p^{2} T^{4}
61C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
67C22C_2^2 1−46T2+p2T4 1 - 46 T^{2} + p^{2} T^{4}
71C22C_2^2 1+97T2+p2T4 1 + 97 T^{2} + p^{2} T^{4}
73C2C_2 (1−14T+pT2)2 ( 1 - 14 T + p T^{2} )^{2}
79C22C_2^2 1+78T2+p2T4 1 + 78 T^{2} + p^{2} T^{4}
83C22C_2^2 1−154T2+p2T4 1 - 154 T^{2} + p^{2} T^{4}
89C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
97C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
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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

−8.515245810440513603401246871378, −8.179973661375063929673139627961, −7.75259560555948641402045641659, −7.67874062083405697403242203637, −7.37186909254632687565611469841, −7.25744860291917568378418205894, −6.42621022266829406806621269248, −6.26172620424009590100880126171, −5.50627003545573358916860676169, −5.39207713387039701470231963984, −4.69922555728763204763305805093, −4.54966527534769299636854543770, −3.95102930778469607942271498610, −3.62155193860924093862598511061, −3.29805567501927992257675175863, −3.21486104860728289780686034821, −2.10768653893582006582456553436, −1.86810220297277951543526585012, −0.77836902662681759368104591465, −0.39758180144335461761380036596, 0.39758180144335461761380036596, 0.77836902662681759368104591465, 1.86810220297277951543526585012, 2.10768653893582006582456553436, 3.21486104860728289780686034821, 3.29805567501927992257675175863, 3.62155193860924093862598511061, 3.95102930778469607942271498610, 4.54966527534769299636854543770, 4.69922555728763204763305805093, 5.39207713387039701470231963984, 5.50627003545573358916860676169, 6.26172620424009590100880126171, 6.42621022266829406806621269248, 7.25744860291917568378418205894, 7.37186909254632687565611469841, 7.67874062083405697403242203637, 7.75259560555948641402045641659, 8.179973661375063929673139627961, 8.515245810440513603401246871378

Graph of the ZZ-function along the critical line