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Properties

Label 4-3240e2-1.1-c0e2-0-10
Degree 44
Conductor 1049760010497600
Sign 11
Analytic cond. 2.614592.61459
Root an. cond. 1.271601.27160
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  + 2-s − 5-s − 8-s − 10-s − 16-s + 2·19-s + 23-s + 3·31-s + 2·38-s + 40-s + 46-s + 2·47-s + 49-s + 2·53-s − 3·61-s + 3·62-s + 64-s + 3·79-s + 80-s − 3·83-s + 2·94-s − 2·95-s + 98-s + 2·106-s − 115-s + 121-s − 3·122-s + ⋯
L(s)  = 1  + 2-s − 5-s − 8-s − 10-s − 16-s + 2·19-s + 23-s + 3·31-s + 2·38-s + 40-s + 46-s + 2·47-s + 49-s + 2·53-s − 3·61-s + 3·62-s + 64-s + 3·79-s + 80-s − 3·83-s + 2·94-s − 2·95-s + 98-s + 2·106-s − 115-s + 121-s − 3·122-s + ⋯

Functional equation

Λ(s)=(10497600s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(10497600s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1049760010497600    =    26⋅38⋅522^{6} \cdot 3^{8} \cdot 5^{2}
Sign: 11
Analytic conductor: 2.614592.61459
Root analytic conductor: 1.271601.27160
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 10497600, ( :0,0), 1)(4,\ 10497600,\ (\ :0, 0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.8294216241.829421624
L(12)L(\frac12) ≈\approx 1.8294216241.829421624
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)
bad2C2C_2 1−T+T2 1 - T + T^{2}
3 1 1
5C2C_2 1+T+T2 1 + T + T^{2}
good7C22C_2^2 1−T2+T4 1 - T^{2} + T^{4}
11C22C_2^2 1−T2+T4 1 - T^{2} + T^{4}
13C22C_2^2 1−T2+T4 1 - T^{2} + T^{4}
17C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
19C2C_2 (1−T+T2)2 ( 1 - T + T^{2} )^{2}
23C1C_1×\timesC2C_2 (1−T)2(1+T+T2) ( 1 - T )^{2}( 1 + T + T^{2} )
29C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
31C1C_1×\timesC2C_2 (1−T)2(1−T+T2) ( 1 - T )^{2}( 1 - T + T^{2} )
37C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
41C22C_2^2 1−T2+T4 1 - T^{2} + T^{4}
43C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
47C2C_2 (1−T+T2)2 ( 1 - T + T^{2} )^{2}
53C2C_2 (1−T+T2)2 ( 1 - T + T^{2} )^{2}
59C22C_2^2 1−T2+T4 1 - T^{2} + T^{4}
61C1C_1×\timesC2C_2 (1+T)2(1+T+T2) ( 1 + T )^{2}( 1 + T + T^{2} )
67C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
71C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
73C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
79C1C_1×\timesC2C_2 (1−T)2(1−T+T2) ( 1 - T )^{2}( 1 - T + T^{2} )
83C1C_1×\timesC2C_2 (1+T)2(1+T+T2) ( 1 + T )^{2}( 1 + T + T^{2} )
89C2C_2 (1+T2)2 ( 1 + T^{2} )^{2}
97C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + 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

−8.872271334701190448041472904085, −8.811558973847702002655833772939, −8.186860105875462530177872551970, −7.83742917416974922249155223262, −7.50192707701956675179681701891, −7.21109525321414142132101978870, −6.72186047110978722716552833667, −6.37538762640902366912605595907, −5.78625466329583375377506267025, −5.63689739952118966535027718403, −5.12441084845364472506472216577, −4.68556014356963110531476053953, −4.44139765229329629851185928633, −3.99993929063249794312464487497, −3.55801591570220950716833100703, −3.17829903633742274743160476213, −2.65501909914820034429238370499, −2.48135939208053609214986728750, −1.21172007572193496484057924199, −0.827728302549680570199567503576, 0.827728302549680570199567503576, 1.21172007572193496484057924199, 2.48135939208053609214986728750, 2.65501909914820034429238370499, 3.17829903633742274743160476213, 3.55801591570220950716833100703, 3.99993929063249794312464487497, 4.44139765229329629851185928633, 4.68556014356963110531476053953, 5.12441084845364472506472216577, 5.63689739952118966535027718403, 5.78625466329583375377506267025, 6.37538762640902366912605595907, 6.72186047110978722716552833667, 7.21109525321414142132101978870, 7.50192707701956675179681701891, 7.83742917416974922249155223262, 8.186860105875462530177872551970, 8.811558973847702002655833772939, 8.872271334701190448041472904085

Graph of the ZZ-function along the critical line