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Properties

Label 4-855e2-1.1-c0e2-0-3
Degree 44
Conductor 731025731025
Sign 11
Analytic cond. 0.1820730.182073
Root an. cond. 0.6532230.653223
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 − 3-s + 4-s − 5-s − 6-s + 2·8-s − 10-s + 11-s − 12-s + 13-s + 15-s + 2·16-s + 2·19-s − 20-s + 22-s − 2·24-s + 26-s + 27-s + 30-s + 2·32-s − 33-s − 2·37-s + 2·38-s − 39-s − 2·40-s + 44-s − 2·48-s + ⋯
L(s)  = 1  + 2-s − 3-s + 4-s − 5-s − 6-s + 2·8-s − 10-s + 11-s − 12-s + 13-s + 15-s + 2·16-s + 2·19-s − 20-s + 22-s − 2·24-s + 26-s + 27-s + 30-s + 2·32-s − 33-s − 2·37-s + 2·38-s − 39-s − 2·40-s + 44-s − 2·48-s + ⋯

Functional equation

Λ(s)=(731025s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(731025s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 731025731025    =    34⋅52⋅1923^{4} \cdot 5^{2} \cdot 19^{2}
Sign: 11
Analytic conductor: 0.1820730.182073
Root analytic conductor: 0.6532230.653223
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 731025, ( :0,0), 1)(4,\ 731025,\ (\ :0, 0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.3218775031.321877503
L(12)L(\frac12) ≈\approx 1.3218775031.321877503
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)
bad3C2C_2 1+T+T2 1 + T + T^{2}
5C2C_2 1+T+T2 1 + T + T^{2}
19C1C_1 (1−T)2 ( 1 - T )^{2}
good2C1C_1×\timesC2C_2 (1−T)2(1+T+T2) ( 1 - T )^{2}( 1 + T + T^{2} )
7C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
11C1C_1×\timesC2C_2 (1−T)2(1+T+T2) ( 1 - T )^{2}( 1 + T + T^{2} )
13C1C_1×\timesC2C_2 (1−T)2(1+T+T2) ( 1 - T )^{2}( 1 + T + T^{2} )
17C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
23C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
29C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
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}
41C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
43C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
47C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
53C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
59C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
61C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
67C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{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}
79C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
83C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
89C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
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

−10.67902765044076885685885639404, −10.62263112617358495073049964707, −9.800251267550109782204396607383, −9.499499425048540020274206872204, −8.871761633891009084291801383350, −8.273782596628958866388757061487, −7.920804388662517541291852012680, −7.52378973501114074844991737809, −7.01066731434009246166094142285, −6.69317900147863393963269333666, −6.25648599734183195164879183440, −5.53196818025767612196188082667, −5.49174128296106319361309973377, −4.67521868356753176431705395629, −4.47188274104729856876338173000, −3.87517981966162602449470147794, −3.22144856179938156064369391772, −3.11011177603211712190796048487, −1.58946467751898220065287823715, −1.35611621016606595567620269397, 1.35611621016606595567620269397, 1.58946467751898220065287823715, 3.11011177603211712190796048487, 3.22144856179938156064369391772, 3.87517981966162602449470147794, 4.47188274104729856876338173000, 4.67521868356753176431705395629, 5.49174128296106319361309973377, 5.53196818025767612196188082667, 6.25648599734183195164879183440, 6.69317900147863393963269333666, 7.01066731434009246166094142285, 7.52378973501114074844991737809, 7.920804388662517541291852012680, 8.273782596628958866388757061487, 8.871761633891009084291801383350, 9.499499425048540020274206872204, 9.800251267550109782204396607383, 10.62263112617358495073049964707, 10.67902765044076885685885639404

Graph of the ZZ-function along the critical line