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Properties

Label 4-8788-1.1-c1e2-0-1
Degree 44
Conductor 87888788
Sign 11
Analytic cond. 0.5603300.560330
Root an. cond. 0.8651890.865189
Motivic weight 11
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  + 2·3-s + 4-s − 3·9-s + 2·12-s + 13-s + 16-s − 6·17-s − 25-s − 14·27-s + 12·29-s − 3·36-s + 2·39-s − 2·43-s + 2·48-s − 13·49-s − 12·51-s + 52-s + 16·61-s + 64-s − 6·68-s − 2·75-s + 16·79-s − 4·81-s + 24·87-s − 100-s − 24·101-s − 8·103-s + ⋯
L(s)  = 1  + 1.15·3-s + 1/2·4-s − 9-s + 0.577·12-s + 0.277·13-s + 1/4·16-s − 1.45·17-s − 1/5·25-s − 2.69·27-s + 2.22·29-s − 1/2·36-s + 0.320·39-s − 0.304·43-s + 0.288·48-s − 1.85·49-s − 1.68·51-s + 0.138·52-s + 2.04·61-s + 1/8·64-s − 0.727·68-s − 0.230·75-s + 1.80·79-s − 4/9·81-s + 2.57·87-s − 0.0999·100-s − 2.38·101-s − 0.788·103-s + ⋯

Functional equation

Λ(s)=(8788s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 8788 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(8788s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 8788 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 87888788    =    22⋅1332^{2} \cdot 13^{3}
Sign: 11
Analytic conductor: 0.5603300.560330
Root analytic conductor: 0.8651890.865189
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 8788, ( :1/2,1/2), 1)(4,\ 8788,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.3270500771.327050077
L(12)L(\frac12) ≈\approx 1.3270500771.327050077
L(32)L(\frac{3}{2}) 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)
bad2C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
13C1C_1 1−T 1 - T
good3C2C_2 (1−T+pT2)2 ( 1 - T + p T^{2} )^{2}
5C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
7C2C_2 (1−T+pT2)(1+T+pT2) ( 1 - T + p T^{2} )( 1 + T + p T^{2} )
11C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
17C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
19C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
29C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
31C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
37C2C_2 (1−7T+pT2)(1+7T+pT2) ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} )
41C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
43C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
47C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
53C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
59C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
61C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
67C2C_2 (1−14T+pT2)(1+14T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} )
71C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
73C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
79C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
83C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
89C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
97C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p 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

−11.52736358384083380205994337321, −11.21486999262721520146961750481, −10.59579744741587591987377046637, −9.848425754886298251834442498391, −9.252086119982469784390910809299, −8.631851122063147597031259612932, −8.289125051595985253760732717123, −7.85100333312502477176899037562, −6.76423132327857768444846951818, −6.43741307804194710670707219872, −5.56442463767257268744193982616, −4.68671862775252250000074194944, −3.64028761626013442697591583469, −2.86630826934752796258245605486, −2.19431989234797768682378556675, 2.19431989234797768682378556675, 2.86630826934752796258245605486, 3.64028761626013442697591583469, 4.68671862775252250000074194944, 5.56442463767257268744193982616, 6.43741307804194710670707219872, 6.76423132327857768444846951818, 7.85100333312502477176899037562, 8.289125051595985253760732717123, 8.631851122063147597031259612932, 9.252086119982469784390910809299, 9.848425754886298251834442498391, 10.59579744741587591987377046637, 11.21486999262721520146961750481, 11.52736358384083380205994337321

Graph of the ZZ-function along the critical line