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Properties

Label 4-1134e2-1.1-c1e2-0-72
Degree 44
Conductor 12859561285956
Sign −1-1
Analytic cond. 81.993681.9936
Root an. cond. 3.009153.00915
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 11

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s + 2·7-s − 4·8-s + 6·11-s − 4·14-s + 5·16-s − 12·22-s + 12·23-s − 10·25-s + 6·28-s − 12·29-s − 6·32-s − 8·37-s − 2·43-s + 18·44-s − 24·46-s − 3·49-s + 20·50-s − 24·53-s − 8·56-s + 24·58-s + 7·64-s + 10·67-s + 24·71-s + 16·74-s + 12·77-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s + 0.755·7-s − 1.41·8-s + 1.80·11-s − 1.06·14-s + 5/4·16-s − 2.55·22-s + 2.50·23-s − 2·25-s + 1.13·28-s − 2.22·29-s − 1.06·32-s − 1.31·37-s − 0.304·43-s + 2.71·44-s − 3.53·46-s − 3/7·49-s + 2.82·50-s − 3.29·53-s − 1.06·56-s + 3.15·58-s + 7/8·64-s + 1.22·67-s + 2.84·71-s + 1.85·74-s + 1.36·77-s + ⋯

Functional equation

Λ(s)=(1285956s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(1285956s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 12859561285956    =    22⋅38⋅722^{2} \cdot 3^{8} \cdot 7^{2}
Sign: −1-1
Analytic conductor: 81.993681.9936
Root analytic conductor: 3.009153.00915
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 1285956, ( :1/2,1/2), −1)(4,\ 1285956,\ (\ :1/2, 1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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 (1+T)2 ( 1 + T )^{2}
3 1 1
7C2C_2 1−2T+pT2 1 - 2 T + p T^{2}
good5C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
11C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
13C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
17C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
19C2C_2 (1−T+pT2)(1+T+pT2) ( 1 - T + p T^{2} )( 1 + T + p T^{2} )
23C2C_2 (1−6T+pT2)2 ( 1 - 6 T + 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+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
41C2C_2 (1−9T+pT2)(1+9T+pT2) ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} )
43C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
47C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
53C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
59C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
61C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
67C2C_2 (1−5T+pT2)2 ( 1 - 5 T + p T^{2} )^{2}
71C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
73C2C_2 (1−11T+pT2)(1+11T+pT2) ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} )
79C2C_2 (1+4T+pT2)2 ( 1 + 4 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 (1−5T+pT2)(1+5T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 5 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

−7.80670854475898716623947697687, −7.57486075433264375877702850521, −6.83856700403258930718083224666, −6.68619713403218599725876916726, −6.35782135459828133998327355434, −5.48525873198846016080664036461, −5.34383976963704470462884839275, −4.67165130826644550690102729752, −3.84523654364808887700815920458, −3.65076060485342096132488694617, −2.98455636915823987690855594247, −2.07379971708251484770168742942, −1.60693251106342657124988453680, −1.22338111669750545882567189127, 0, 1.22338111669750545882567189127, 1.60693251106342657124988453680, 2.07379971708251484770168742942, 2.98455636915823987690855594247, 3.65076060485342096132488694617, 3.84523654364808887700815920458, 4.67165130826644550690102729752, 5.34383976963704470462884839275, 5.48525873198846016080664036461, 6.35782135459828133998327355434, 6.68619713403218599725876916726, 6.83856700403258930718083224666, 7.57486075433264375877702850521, 7.80670854475898716623947697687

Graph of the ZZ-function along the critical line