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Properties

Label 4-1200e2-1.1-c3e2-0-18
Degree 44
Conductor 14400001440000
Sign 11
Analytic cond. 5012.965012.96
Root an. cond. 8.414408.41440
Motivic weight 33
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  − 9·9-s − 144·11-s + 104·19-s + 156·29-s − 240·31-s + 724·41-s + 670·49-s + 1.39e3·59-s + 444·61-s − 192·71-s − 1.26e3·79-s + 81·81-s − 1.98e3·89-s + 1.29e3·99-s + 1.78e3·101-s − 892·109-s + 1.28e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.35e3·169-s + ⋯
L(s)  = 1  − 1/3·9-s − 3.94·11-s + 1.25·19-s + 0.998·29-s − 1.39·31-s + 2.75·41-s + 1.95·49-s + 3.07·59-s + 0.931·61-s − 0.320·71-s − 1.80·79-s + 1/9·81-s − 2.36·89-s + 1.31·99-s + 1.75·101-s − 0.783·109-s + 9.68·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 1.98·169-s + ⋯

Functional equation

Λ(s)=(1440000s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1440000s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 14400001440000    =    28⋅32⋅542^{8} \cdot 3^{2} \cdot 5^{4}
Sign: 11
Analytic conductor: 5012.965012.96
Root analytic conductor: 8.414408.41440
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 1440000, ( :3/2,3/2), 1)(4,\ 1440000,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 2.2642933662.264293366
L(12)L(\frac12) ≈\approx 2.2642933662.264293366
L(52)L(\frac{5}{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+p2T2 1 + p^{2} T^{2}
5 1 1
good7C22C_2^2 1−670T2+p6T4 1 - 670 T^{2} + p^{6} T^{4}
11C2C_2 (1+72T+p3T2)2 ( 1 + 72 T + p^{3} T^{2} )^{2}
13C22C_2^2 1−4358T2+p6T4 1 - 4358 T^{2} + p^{6} T^{4}
17C22C_2^2 1−8382T2+p6T4 1 - 8382 T^{2} + p^{6} T^{4}
19C2C_2 (1−52T+p3T2)2 ( 1 - 52 T + p^{3} T^{2} )^{2}
23C22C_2^2 1−1230T2+p6T4 1 - 1230 T^{2} + p^{6} T^{4}
29C2C_2 (1−78T+p3T2)2 ( 1 - 78 T + p^{3} T^{2} )^{2}
31C2C_2 (1+120T+p3T2)2 ( 1 + 120 T + p^{3} T^{2} )^{2}
37C22C_2^2 1−78806T2+p6T4 1 - 78806 T^{2} + p^{6} T^{4}
41C2C_2 (1−362T+p3T2)2 ( 1 - 362 T + p^{3} T^{2} )^{2}
43C22C_2^2 1+75242T2+p6T4 1 + 75242 T^{2} + p^{6} T^{4}
47C22C_2^2 1−129246T2+p6T4 1 - 129246 T^{2} + p^{6} T^{4}
53C22C_2^2 1+151146T2+p6T4 1 + 151146 T^{2} + p^{6} T^{4}
59C2C_2 (1−696T+p3T2)2 ( 1 - 696 T + p^{3} T^{2} )^{2}
61C2C_2 (1−222T+p3T2)2 ( 1 - 222 T + p^{3} T^{2} )^{2}
67C22C_2^2 1−601510T2+p6T4 1 - 601510 T^{2} + p^{6} T^{4}
71C2C_2 (1+96T+p3T2)2 ( 1 + 96 T + p^{3} T^{2} )^{2}
73C22C_2^2 1−746350T2+p6T4 1 - 746350 T^{2} + p^{6} T^{4}
79C2C_2 (1+8pT+p3T2)2 ( 1 + 8 p T + p^{3} T^{2} )^{2}
83C22C_2^2 1−769030T2+p6T4 1 - 769030 T^{2} + p^{6} T^{4}
89C2C_2 (1+994T+p3T2)2 ( 1 + 994 T + p^{3} T^{2} )^{2}
97C22C_2^2 1+844610T2+p6T4 1 + 844610 T^{2} + p^{6} 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

−9.609066596854010172545528747345, −9.255258350569777277372392849833, −8.533539371017494799688240676265, −8.349193388379483157296106515950, −7.953878536801475725799277886176, −7.53426842894149902147168880608, −7.14622652905843389522028824439, −7.03325660006936345752057073369, −5.85056114457770477083531204156, −5.70344186151030391049855193375, −5.38922975006984331840333063816, −5.14926164563545981143445552410, −4.40758048122428180016047963412, −4.02672595316898702116303425886, −3.03503269413336474233032630079, −2.90721941915083906025188167616, −2.43971861604042363296237140971, −1.97112195424329039407859023673, −0.61428967038398093731425600990, −0.60206523867416818480962169817, 0.60206523867416818480962169817, 0.61428967038398093731425600990, 1.97112195424329039407859023673, 2.43971861604042363296237140971, 2.90721941915083906025188167616, 3.03503269413336474233032630079, 4.02672595316898702116303425886, 4.40758048122428180016047963412, 5.14926164563545981143445552410, 5.38922975006984331840333063816, 5.70344186151030391049855193375, 5.85056114457770477083531204156, 7.03325660006936345752057073369, 7.14622652905843389522028824439, 7.53426842894149902147168880608, 7.953878536801475725799277886176, 8.349193388379483157296106515950, 8.533539371017494799688240676265, 9.255258350569777277372392849833, 9.609066596854010172545528747345

Graph of the ZZ-function along the critical line