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Properties

Label 4-2736e2-1.1-c0e2-0-3
Degree 44
Conductor 74856967485696
Sign 11
Analytic cond. 1.864431.86443
Root an. cond. 1.168521.16852
Motivic weight 00
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 13-s + 2·19-s + 25-s − 2·37-s + 3·43-s − 49-s + 61-s − 3·67-s − 73-s − 3·79-s + 2·97-s + 2·109-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  − 13-s + 2·19-s + 25-s − 2·37-s + 3·43-s − 49-s + 61-s − 3·67-s − 73-s − 3·79-s + 2·97-s + 2·109-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + 191-s + ⋯

Functional equation

Λ(s)=(7485696s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(7485696s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 74856967485696    =    28⋅34⋅1922^{8} \cdot 3^{4} \cdot 19^{2}
Sign: 11
Analytic conductor: 1.864431.86443
Root analytic conductor: 1.168521.16852
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 7485696, ( :0,0), 1)(4,\ 7485696,\ (\ :0, 0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.2968673861.296867386
L(12)L(\frac12) ≈\approx 1.2968673861.296867386
L(1)L(1) 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
3 1 1
19C1C_1 (1−T)2 ( 1 - T )^{2}
good5C22C_2^2 1−T2+T4 1 - T^{2} + T^{4}
7C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
11C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
13C1C_1×\timesC2C_2 (1+T)2(1−T+T2) ( 1 + T )^{2}( 1 - T + T^{2} )
17C22C_2^2 1−T2+T4 1 - T^{2} + T^{4}
23C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
29C22C_2^2 1−T2+T4 1 - T^{2} + T^{4}
31C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
37C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
41C22C_2^2 1−T2+T4 1 - T^{2} + T^{4}
43C1C_1×\timesC2C_2 (1−T)2(1−T+T2) ( 1 - T )^{2}( 1 - T + T^{2} )
47C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
53C22C_2^2 1−T2+T4 1 - T^{2} + T^{4}
59C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
61C1C_1×\timesC2C_2 (1−T)2(1+T+T2) ( 1 - T )^{2}( 1 + T + T^{2} )
67C1C_1×\timesC2C_2 (1+T)2(1+T+T2) ( 1 + T )^{2}( 1 + T + T^{2} )
71C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
73C1C_1×\timesC2C_2 (1+T)2(1−T+T2) ( 1 + T )^{2}( 1 - T + T^{2} )
79C1C_1×\timesC2C_2 (1+T)2(1+T+T2) ( 1 + T )^{2}( 1 + T + T^{2} )
83C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
89C22C_2^2 1−T2+T4 1 - T^{2} + T^{4}
97C2C_2 (1−T+T2)2 ( 1 - T + 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

−8.952910772874176106280888074193, −8.943994934304339261617236592804, −8.655110398597480460679181888037, −7.87273262613030846417016401448, −7.63823020565407308784191310789, −7.34073364062009789488295651085, −6.96516226400799493069668125762, −6.77888406630811769064961417941, −5.83912396319886297117849941603, −5.79609977516594648501204724341, −5.46945916640562172902027589448, −4.70815618028868880075134397474, −4.66579629539841551547516813598, −4.17810278864933120319254758079, −3.31808992649296495843495233733, −3.18985694617474133857687924156, −2.76213216514709892984274457454, −2.05880563793082294983593070623, −1.50606362395437365325097187522, −0.76959534355878156740201573646, 0.76959534355878156740201573646, 1.50606362395437365325097187522, 2.05880563793082294983593070623, 2.76213216514709892984274457454, 3.18985694617474133857687924156, 3.31808992649296495843495233733, 4.17810278864933120319254758079, 4.66579629539841551547516813598, 4.70815618028868880075134397474, 5.46945916640562172902027589448, 5.79609977516594648501204724341, 5.83912396319886297117849941603, 6.77888406630811769064961417941, 6.96516226400799493069668125762, 7.34073364062009789488295651085, 7.63823020565407308784191310789, 7.87273262613030846417016401448, 8.655110398597480460679181888037, 8.943994934304339261617236592804, 8.952910772874176106280888074193

Graph of the ZZ-function along the critical line