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Properties

Label 4-3344e2-1.1-c0e2-0-2
Degree 44
Conductor 1118233611182336
Sign 11
Analytic cond. 2.785132.78513
Root an. cond. 1.291841.29184
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  − 5-s + 2·7-s + 9-s − 2·11-s − 19-s + 25-s − 2·35-s + 2·43-s − 45-s + 49-s − 3·53-s + 2·55-s + 2·63-s − 4·77-s − 3·79-s + 2·83-s + 95-s + 3·97-s − 2·99-s + 3·121-s − 2·125-s + 127-s + 131-s − 2·133-s + 137-s + 139-s + 149-s + ⋯
L(s)  = 1  − 5-s + 2·7-s + 9-s − 2·11-s − 19-s + 25-s − 2·35-s + 2·43-s − 45-s + 49-s − 3·53-s + 2·55-s + 2·63-s − 4·77-s − 3·79-s + 2·83-s + 95-s + 3·97-s − 2·99-s + 3·121-s − 2·125-s + 127-s + 131-s − 2·133-s + 137-s + 139-s + 149-s + ⋯

Functional equation

Λ(s)=(11182336s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 11182336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(11182336s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 11182336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1118233611182336    =    28⋅112⋅1922^{8} \cdot 11^{2} \cdot 19^{2}
Sign: 11
Analytic conductor: 2.785132.78513
Root analytic conductor: 1.291841.29184
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 11182336, ( :0,0), 1)(4,\ 11182336,\ (\ :0, 0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.2467778531.246777853
L(12)L(\frac12) ≈\approx 1.2467778531.246777853
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
11C1C_1 (1+T)2 ( 1 + T )^{2}
19C2C_2 1+T+T2 1 + T + T^{2}
good3C22C_2^2 1−T2+T4 1 - T^{2} + T^{4}
5C1C_1×\timesC2C_2 (1+T)2(1−T+T2) ( 1 + T )^{2}( 1 - T + T^{2} )
7C2C_2 (1−T+T2)2 ( 1 - T + T^{2} )^{2}
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} )
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+T2)2 ( 1 + T^{2} )^{2}
37C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
41C22C_2^2 1−T2+T4 1 - T^{2} + T^{4}
43C2C_2 (1−T+T2)2 ( 1 - T + T^{2} )^{2}
47C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
53C1C_1×\timesC2C_2 (1+T)2(1+T+T2) ( 1 + T )^{2}( 1 + T + T^{2} )
59C22C_2^2 1−T2+T4 1 - T^{2} + T^{4}
61C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
67C22C_2^2 1−T2+T4 1 - T^{2} + T^{4}
71C22C_2^2 1−T2+T4 1 - T^{2} + T^{4}
73C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
79C1C_1×\timesC2C_2 (1+T)2(1+T+T2) ( 1 + T )^{2}( 1 + T + T^{2} )
83C2C_2 (1−T+T2)2 ( 1 - T + T^{2} )^{2}
89C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
97C1C_1×\timesC2C_2 (1−T)2(1−T+T2) ( 1 - 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

−8.794732558489193085887053209647, −8.557382882040194823215745021452, −7.959388314512843262717855707739, −7.911448115160538493617567579201, −7.71451313757838681602529753278, −7.32229208299773081309552172946, −6.96093219636892474800478845206, −6.35300844636509689178891268290, −5.97948721888090944897322787390, −5.41885627377976141548132539303, −5.06481516689991472056421212404, −4.63180437365456102900140970278, −4.49243754141138424659596183661, −4.22541799328451635045985275635, −3.47166690818274085829073388825, −3.02058738725274995818374435469, −2.50489588765767424931020944921, −1.86901931898415632542501174022, −1.62089124365510548962845749433, −0.67650917319613382229966624947, 0.67650917319613382229966624947, 1.62089124365510548962845749433, 1.86901931898415632542501174022, 2.50489588765767424931020944921, 3.02058738725274995818374435469, 3.47166690818274085829073388825, 4.22541799328451635045985275635, 4.49243754141138424659596183661, 4.63180437365456102900140970278, 5.06481516689991472056421212404, 5.41885627377976141548132539303, 5.97948721888090944897322787390, 6.35300844636509689178891268290, 6.96093219636892474800478845206, 7.32229208299773081309552172946, 7.71451313757838681602529753278, 7.911448115160538493617567579201, 7.959388314512843262717855707739, 8.557382882040194823215745021452, 8.794732558489193085887053209647

Graph of the ZZ-function along the critical line