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Properties

Label 4-700e2-1.1-c1e2-0-3
Degree 44
Conductor 490000490000
Sign 11
Analytic cond. 31.242831.2428
Root an. cond. 2.364212.36421
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s + 3·9-s − 4·16-s − 6·18-s + 2·29-s + 8·32-s + 6·36-s − 7·49-s + 20·53-s − 4·58-s − 8·64-s + 14·98-s − 40·106-s + 30·109-s − 16·113-s + 4·116-s − 21·121-s + 127-s + 131-s + 137-s + 139-s − 12·144-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s + 9-s − 16-s − 1.41·18-s + 0.371·29-s + 1.41·32-s + 36-s − 49-s + 2.74·53-s − 0.525·58-s − 64-s + 1.41·98-s − 3.88·106-s + 2.87·109-s − 1.50·113-s + 0.371·116-s − 1.90·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯

Functional equation

Λ(s)=(490000s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(490000s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 490000490000    =    24⋅54⋅722^{4} \cdot 5^{4} \cdot 7^{2}
Sign: 11
Analytic conductor: 31.242831.2428
Root analytic conductor: 2.364212.36421
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 490000, ( :1/2,1/2), 1)(4,\ 490000,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.92056368960.9205636896
L(12)L(\frac12) ≈\approx 0.92056368960.9205636896
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)
bad2C2C_2 1+pT+pT2 1 + p T + p T^{2}
5 1 1
7C2C_2 1+pT2 1 + p T^{2}
good3C2C_2 (1−pT+pT2)(1+pT+pT2) ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} )
11C2C_2 (1−T+pT2)(1+T+pT2) ( 1 - T + p T^{2} )( 1 + T + p T^{2} )
13C22C_2^2 1−17T2+p2T4 1 - 17 T^{2} + p^{2} T^{4}
17C22C_2^2 1−25T2+p2T4 1 - 25 T^{2} + p^{2} T^{4}
19C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
23C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
29C2C_2 (1−T+pT2)2 ( 1 - T + p T^{2} )^{2}
31C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
37C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
41C22C_2^2 1−46T2+p2T4 1 - 46 T^{2} + p^{2} T^{4}
43C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
47C2C_2 (1−9T+pT2)(1+9T+pT2) ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} )
53C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
59C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
61C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
67C2C_2 (1−14T+pT2)(1+14T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} )
71C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
73C2C_2 (1−16T+pT2)(1+16T+pT2) ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} )
79C2C_2 (1−T+pT2)(1+T+pT2) ( 1 - T + p T^{2} )( 1 + T + p T^{2} )
83C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
89C22C_2^2 1−34T2+p2T4 1 - 34 T^{2} + p^{2} T^{4}
97C22C_2^2 1+31T2+p2T4 1 + 31 T^{2} + 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

−8.399547694591570114395175387242, −8.304482940675401206155783925314, −7.64024279260154245455752826177, −7.26567967929252176925762595750, −6.93095569414943302325531295146, −6.48804258116661570681810866779, −5.87671722293011809683501951143, −5.26166490985109088140711756770, −4.67263530229777170111796408991, −4.21436753526871820554033625305, −3.65340200055647704790741625600, −2.80747401560655361004950373148, −2.12076040886554542994327127542, −1.47583381429041937688787004072, −0.68533387951976935347434466916, 0.68533387951976935347434466916, 1.47583381429041937688787004072, 2.12076040886554542994327127542, 2.80747401560655361004950373148, 3.65340200055647704790741625600, 4.21436753526871820554033625305, 4.67263530229777170111796408991, 5.26166490985109088140711756770, 5.87671722293011809683501951143, 6.48804258116661570681810866779, 6.93095569414943302325531295146, 7.26567967929252176925762595750, 7.64024279260154245455752826177, 8.304482940675401206155783925314, 8.399547694591570114395175387242

Graph of the ZZ-function along the critical line