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Properties

Label 4-5376e2-1.1-c1e2-0-46
Degree 44
Conductor 2890137628901376
Sign 11
Analytic cond. 1842.771842.77
Root an. cond. 6.551916.55191
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·7-s − 9-s − 12·23-s + 10·25-s − 16·31-s − 24·41-s − 24·47-s + 3·49-s − 2·63-s + 12·71-s + 20·73-s + 8·79-s + 81-s − 24·89-s − 20·97-s + 16·103-s − 12·113-s − 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 24·161-s + 163-s + ⋯
L(s)  = 1  + 0.755·7-s − 1/3·9-s − 2.50·23-s + 2·25-s − 2.87·31-s − 3.74·41-s − 3.50·47-s + 3/7·49-s − 0.251·63-s + 1.42·71-s + 2.34·73-s + 0.900·79-s + 1/9·81-s − 2.54·89-s − 2.03·97-s + 1.57·103-s − 1.12·113-s − 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s − 1.89·161-s + 0.0783·163-s + ⋯

Functional equation

Λ(s)=(28901376s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 28901376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(28901376s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 28901376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 2890137628901376    =    216⋅32⋅722^{16} \cdot 3^{2} \cdot 7^{2}
Sign: 11
Analytic conductor: 1842.771842.77
Root analytic conductor: 6.551916.55191
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 28901376, ( :1/2,1/2), 1)(4,\ 28901376,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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+T2 1 + T^{2}
7C1C_1 (1−T)2 ( 1 - T )^{2}
good5C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
11C22C_2^2 1+14T2+p2T4 1 + 14 T^{2} + p^{2} T^{4}
13C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
17C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
19C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
23C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
29C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
31C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
37C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
41C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
43C22C_2^2 1−70T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
47C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
53C22C_2^2 1−70T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
59C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
61C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
67C22C_2^2 1−70T2+p2T4 1 - 70 T^{2} + p^{2} T^{4}
71C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
73C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
79C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
83C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
89C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
97C2C_2 (1+10T+pT2)2 ( 1 + 10 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

−7.983081919844388091019770191939, −7.942696774340037832322275123907, −7.16871189149650957205261704801, −6.89595211369618080201968592524, −6.60145928989985901456930619813, −6.39204791689497160412699890303, −5.57844285564960434267849504143, −5.54229606706781697400865913072, −5.02742272292131235884624338243, −4.87548881225291001669457992762, −4.37511585051108608285918087555, −3.72163334974701637072978281481, −3.46445598051237037944234230449, −3.32011788639021068020984106117, −2.44472394532890852593298668173, −2.06875207797482378266165752435, −1.64127100441604991309627763858, −1.29227534929850042669879402250, 0, 0, 1.29227534929850042669879402250, 1.64127100441604991309627763858, 2.06875207797482378266165752435, 2.44472394532890852593298668173, 3.32011788639021068020984106117, 3.46445598051237037944234230449, 3.72163334974701637072978281481, 4.37511585051108608285918087555, 4.87548881225291001669457992762, 5.02742272292131235884624338243, 5.54229606706781697400865913072, 5.57844285564960434267849504143, 6.39204791689497160412699890303, 6.60145928989985901456930619813, 6.89595211369618080201968592524, 7.16871189149650957205261704801, 7.942696774340037832322275123907, 7.983081919844388091019770191939

Graph of the ZZ-function along the critical line