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Properties

Label 4-1095200-1.1-c1e2-0-7
Degree 44
Conductor 10952001095200
Sign 11
Analytic cond. 69.830969.8309
Root an. cond. 2.890752.89075
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-s + 4-s + 2·5-s + 8-s + 2·9-s + 2·10-s − 4·13-s + 16-s + 8·17-s + 2·18-s + 2·20-s + 3·25-s − 4·26-s − 8·29-s + 32-s + 8·34-s + 2·36-s + 2·37-s + 2·40-s + 12·41-s + 4·45-s + 6·49-s + 3·50-s − 4·52-s − 8·58-s + 64-s − 8·65-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.353·8-s + 2/3·9-s + 0.632·10-s − 1.10·13-s + 1/4·16-s + 1.94·17-s + 0.471·18-s + 0.447·20-s + 3/5·25-s − 0.784·26-s − 1.48·29-s + 0.176·32-s + 1.37·34-s + 1/3·36-s + 0.328·37-s + 0.316·40-s + 1.87·41-s + 0.596·45-s + 6/7·49-s + 0.424·50-s − 0.554·52-s − 1.05·58-s + 1/8·64-s − 0.992·65-s + ⋯

Functional equation

Λ(s)=(1095200s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1095200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1095200s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1095200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 10952001095200    =    25⋅52⋅3722^{5} \cdot 5^{2} \cdot 37^{2}
Sign: 11
Analytic conductor: 69.830969.8309
Root analytic conductor: 2.890752.89075
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 1095200, ( :1/2,1/2), 1)(4,\ 1095200,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 4.5567944234.556794423
L(12)L(\frac12) ≈\approx 4.5567944234.556794423
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)
bad2C1C_1 1−T 1 - T
5C1C_1 (1−T)2 ( 1 - T )^{2}
37C1C_1 (1−T)2 ( 1 - T )^{2}
good3C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
7C22C_2^2 1−6T2+p2T4 1 - 6 T^{2} + p^{2} T^{4}
11C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
13C2C_2×\timesC2C_2 (1−2T+pT2)(1+6T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} )
17C2C_2×\timesC2C_2 (1−6T+pT2)(1−2T+pT2) ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} )
19C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 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×\timesC2C_2 (1+2T+pT2)(1+6T+pT2) ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} )
31C22C_2^2 1−50T2+p2T4 1 - 50 T^{2} + p^{2} T^{4}
41C2C_2×\timesC2C_2 (1−10T+pT2)(1−2T+pT2) ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} )
43C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
47C22C_2^2 1−30T2+p2T4 1 - 30 T^{2} + p^{2} T^{4}
53C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
59C22C_2^2 1−70T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
61C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
67C22C_2^2 1−26T2+p2T4 1 - 26 T^{2} + p^{2} T^{4}
71C22C_2^2 1+62T2+p2T4 1 + 62 T^{2} + p^{2} T^{4}
73C2C_2×\timesC2C_2 (1−10T+pT2)(1+6T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} )
79C22C_2^2 1−106T2+p2T4 1 - 106 T^{2} + p^{2} T^{4}
83C22C_2^2 1−82T2+p2T4 1 - 82 T^{2} + p^{2} T^{4}
89C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
97C2C_2×\timesC2C_2 (1−2T+pT2)(1+14T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{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

−7.80021022775430441531408809226, −7.58585527266948775876736657933, −7.20552315909597070533843902842, −6.78427025973501634097775334182, −6.15946106985185681748568436970, −5.67352503947890013636487329766, −5.51832655236226650629974468909, −5.04167159885489444977418212712, −4.37479970783640735547939347351, −4.04847985564756178140577258055, −3.35555804325760842479407023818, −2.83908832562660018083042083258, −2.26278253704023129981768029003, −1.66833694381427609691398347883, −0.905375457934824433129038326127, 0.905375457934824433129038326127, 1.66833694381427609691398347883, 2.26278253704023129981768029003, 2.83908832562660018083042083258, 3.35555804325760842479407023818, 4.04847985564756178140577258055, 4.37479970783640735547939347351, 5.04167159885489444977418212712, 5.51832655236226650629974468909, 5.67352503947890013636487329766, 6.15946106985185681748568436970, 6.78427025973501634097775334182, 7.20552315909597070533843902842, 7.58585527266948775876736657933, 7.80021022775430441531408809226

Graph of the ZZ-function along the critical line