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Properties

Label 4-525e2-1.1-c0e2-0-1
Degree 44
Conductor 275625275625
Sign 11
Analytic cond. 0.06864870.0686487
Root an. cond. 0.5118680.511868
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  + 3-s − 4-s + 7-s − 12-s + 2·13-s + 19-s + 21-s − 27-s − 28-s − 2·31-s − 37-s + 2·39-s − 4·43-s − 2·52-s + 57-s + 61-s + 64-s − 67-s − 73-s − 76-s + 79-s − 81-s − 84-s + 2·91-s − 2·93-s + 2·97-s − 103-s + ⋯
L(s)  = 1  + 3-s − 4-s + 7-s − 12-s + 2·13-s + 19-s + 21-s − 27-s − 28-s − 2·31-s − 37-s + 2·39-s − 4·43-s − 2·52-s + 57-s + 61-s + 64-s − 67-s − 73-s − 76-s + 79-s − 81-s − 84-s + 2·91-s − 2·93-s + 2·97-s − 103-s + ⋯

Functional equation

Λ(s)=(275625s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(275625s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 275625275625    =    32⋅54⋅723^{2} \cdot 5^{4} \cdot 7^{2}
Sign: 11
Analytic conductor: 0.06864870.0686487
Root analytic conductor: 0.5118680.511868
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 275625, ( :0,0), 1)(4,\ 275625,\ (\ :0, 0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.94929074960.9492907496
L(12)L(\frac12) ≈\approx 0.94929074960.9492907496
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}
5 1 1
7C2C_2 1−T+T2 1 - T + T^{2}
good2C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
11C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
13C2C_2 (1−T+T2)2 ( 1 - T + T^{2} )^{2}
17C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
19C1C_1×\timesC2C_2 (1−T)2(1+T+T2) ( 1 - T )^{2}( 1 + T + T^{2} )
23C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
29C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
31C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
37C1C_1×\timesC2C_2 (1+T)2(1−T+T2) ( 1 + T )^{2}( 1 - T + T^{2} )
41C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
43C1C_1 (1+T)4 ( 1 + T )^{4}
47C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
53C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
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} )
71C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + 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}
89C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + 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

−11.16540106269567765907264127829, −11.07924151792112251502279409422, −10.16007332407537852246283458413, −10.04655475174635568119719741655, −9.297326645043563784764588132241, −8.824183963331549155982399384411, −8.788553050784162048287542945783, −8.333473784875796818150507635781, −7.912143297160387157663566225452, −7.46177816708211098543208681219, −6.79320098900243615942595236088, −6.29405059514878773586456937238, −5.38076380476054149900106785153, −5.36167476301944528068221235238, −4.66579447951206809222453269216, −3.95724525056339834796238152931, −3.43586964224177625454046849980, −3.25564162382170338486175477751, −1.95026617534710363294073473319, −1.51510240631486012590937124659, 1.51510240631486012590937124659, 1.95026617534710363294073473319, 3.25564162382170338486175477751, 3.43586964224177625454046849980, 3.95724525056339834796238152931, 4.66579447951206809222453269216, 5.36167476301944528068221235238, 5.38076380476054149900106785153, 6.29405059514878773586456937238, 6.79320098900243615942595236088, 7.46177816708211098543208681219, 7.912143297160387157663566225452, 8.333473784875796818150507635781, 8.788553050784162048287542945783, 8.824183963331549155982399384411, 9.297326645043563784764588132241, 10.04655475174635568119719741655, 10.16007332407537852246283458413, 11.07924151792112251502279409422, 11.16540106269567765907264127829

Graph of the ZZ-function along the critical line