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Properties

Label 4-1440e2-1.1-c1e2-0-48
Degree 44
Conductor 20736002073600
Sign 11
Analytic cond. 132.214132.214
Root an. cond. 3.390933.39093
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  + 3·3-s + 5-s + 4·7-s + 6·9-s − 11-s + 3·15-s − 2·17-s + 14·19-s + 12·21-s − 6·23-s + 9·27-s + 2·29-s + 10·31-s − 3·33-s + 4·35-s + 16·37-s − 5·41-s − 5·43-s + 6·45-s + 4·47-s + 7·49-s − 6·51-s − 20·53-s − 55-s + 42·57-s + 9·59-s + 10·61-s + ⋯
L(s)  = 1  + 1.73·3-s + 0.447·5-s + 1.51·7-s + 2·9-s − 0.301·11-s + 0.774·15-s − 0.485·17-s + 3.21·19-s + 2.61·21-s − 1.25·23-s + 1.73·27-s + 0.371·29-s + 1.79·31-s − 0.522·33-s + 0.676·35-s + 2.63·37-s − 0.780·41-s − 0.762·43-s + 0.894·45-s + 0.583·47-s + 49-s − 0.840·51-s − 2.74·53-s − 0.134·55-s + 5.56·57-s + 1.17·59-s + 1.28·61-s + ⋯

Functional equation

Λ(s)=(2073600s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2073600s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 20736002073600    =    210⋅34⋅522^{10} \cdot 3^{4} \cdot 5^{2}
Sign: 11
Analytic conductor: 132.214132.214
Root analytic conductor: 3.390933.39093
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 2073600, ( :1/2,1/2), 1)(4,\ 2073600,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 7.2401547387.240154738
L(12)L(\frac12) ≈\approx 7.2401547387.240154738
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
3C2C_2 1−pT+pT2 1 - p T + p T^{2}
5C2C_2 1−T+T2 1 - T + T^{2}
good7C2C_2 (1−5T+pT2)(1+T+pT2) ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} )
11C22C_2^2 1+T−10T2+pT3+p2T4 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4}
13C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
17C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
19C2C_2 (1−7T+pT2)2 ( 1 - 7 T + p T^{2} )^{2}
23C22C_2^2 1+6T+13T2+6pT3+p2T4 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4}
29C22C_2^2 1−2T−25T2−2pT3+p2T4 1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4}
31C22C_2^2 1−10T+69T2−10pT3+p2T4 1 - 10 T + 69 T^{2} - 10 p T^{3} + p^{2} T^{4}
37C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
41C22C_2^2 1+5T−16T2+5pT3+p2T4 1 + 5 T - 16 T^{2} + 5 p T^{3} + p^{2} T^{4}
43C2C_2 (1−8T+pT2)(1+13T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} )
47C22C_2^2 1−4T−31T2−4pT3+p2T4 1 - 4 T - 31 T^{2} - 4 p T^{3} + p^{2} T^{4}
53C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
59C22C_2^2 1−9T+22T2−9pT3+p2T4 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4}
61C22C_2^2 1−10T+39T2−10pT3+p2T4 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4}
67C22C_2^2 1+3T−58T2+3pT3+p2T4 1 + 3 T - 58 T^{2} + 3 p T^{3} + p^{2} T^{4}
71C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
73C2C_2 (1+11T+pT2)2 ( 1 + 11 T + p T^{2} )^{2}
79C22C_2^2 1−10T+21T2−10pT3+p2T4 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4}
83C22C_2^2 1+4T−67T2+4pT3+p2T4 1 + 4 T - 67 T^{2} + 4 p T^{3} + p^{2} T^{4}
89C2C_2 (1+18T+pT2)2 ( 1 + 18 T + p T^{2} )^{2}
97C22C_2^2 1+7T−48T2+7pT3+p2T4 1 + 7 T - 48 T^{2} + 7 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

−9.798351062923126448121131704690, −9.535848300184255122605227320524, −8.712148427351964475771060359653, −8.508959988498463799152126065969, −8.087847702906179405575879192321, −7.917147713502018798668098894394, −7.37913611746541974496243430497, −7.30198898408694589929224545003, −6.51045819457831156908692603978, −6.06955489432897134233643096259, −5.46459458287792361184142916905, −5.08613778708341914565026245301, −4.45729376076612660939166826063, −4.39974970225123159878759651307, −3.53825745495048976239772471053, −3.13773298847982114725143311162, −2.55526421771305718533387832471, −2.28504155336601607978579438023, −1.28364590297471768186510535734, −1.27146389956915775190390834897, 1.27146389956915775190390834897, 1.28364590297471768186510535734, 2.28504155336601607978579438023, 2.55526421771305718533387832471, 3.13773298847982114725143311162, 3.53825745495048976239772471053, 4.39974970225123159878759651307, 4.45729376076612660939166826063, 5.08613778708341914565026245301, 5.46459458287792361184142916905, 6.06955489432897134233643096259, 6.51045819457831156908692603978, 7.30198898408694589929224545003, 7.37913611746541974496243430497, 7.917147713502018798668098894394, 8.087847702906179405575879192321, 8.508959988498463799152126065969, 8.712148427351964475771060359653, 9.535848300184255122605227320524, 9.798351062923126448121131704690

Graph of the ZZ-function along the critical line