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Properties

Label 4-312e2-1.1-c1e2-0-20
Degree 44
Conductor 9734497344
Sign 11
Analytic cond. 6.206736.20673
Root an. cond. 1.578391.57839
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 + 3·9-s + 6·13-s − 12·17-s + 16·23-s − 6·25-s + 4·27-s + 12·29-s + 12·39-s − 8·43-s + 10·49-s − 24·51-s − 20·53-s − 4·61-s + 32·69-s − 12·75-s + 5·81-s + 24·87-s + 12·101-s + 16·103-s − 8·107-s − 12·113-s + 18·117-s + 18·121-s + 127-s − 16·129-s + 131-s + ⋯
L(s)  = 1  + 1.15·3-s + 9-s + 1.66·13-s − 2.91·17-s + 3.33·23-s − 6/5·25-s + 0.769·27-s + 2.22·29-s + 1.92·39-s − 1.21·43-s + 10/7·49-s − 3.36·51-s − 2.74·53-s − 0.512·61-s + 3.85·69-s − 1.38·75-s + 5/9·81-s + 2.57·87-s + 1.19·101-s + 1.57·103-s − 0.773·107-s − 1.12·113-s + 1.66·117-s + 1.63·121-s + 0.0887·127-s − 1.40·129-s + 0.0873·131-s + ⋯

Functional equation

Λ(s)=(97344s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(97344s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 9734497344    =    26⋅32⋅1322^{6} \cdot 3^{2} \cdot 13^{2}
Sign: 11
Analytic conductor: 6.206736.20673
Root analytic conductor: 1.578391.57839
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 97344, ( :1/2,1/2), 1)(4,\ 97344,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.4322498222.432249822
L(12)L(\frac12) ≈\approx 2.4322498222.432249822
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)
bad2 1 1
3C1C_1 (1−T)2 ( 1 - T )^{2}
13C2C_2 1−6T+pT2 1 - 6 T + p T^{2}
good5C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
7C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
11C22C_2^2 1−18T2+p2T4 1 - 18 T^{2} + p^{2} T^{4}
17C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
19C22C_2^2 1−34T2+p2T4 1 - 34 T^{2} + p^{2} T^{4}
23C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
29C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
31C22C_2^2 1+38T2+p2T4 1 + 38 T^{2} + p^{2} T^{4}
37C22C_2^2 1−58T2+p2T4 1 - 58 T^{2} + p^{2} T^{4}
41C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
43C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
47C22C_2^2 1−90T2+p2T4 1 - 90 T^{2} + p^{2} T^{4}
53C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
59C22C_2^2 1+78T2+p2T4 1 + 78 T^{2} + p^{2} T^{4}
61C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
67C22C_2^2 1−130T2+p2T4 1 - 130 T^{2} + p^{2} T^{4}
71C22C_2^2 1−106T2+p2T4 1 - 106 T^{2} + p^{2} T^{4}
73C22C_2^2 1−82T2+p2T4 1 - 82 T^{2} + p^{2} T^{4}
79C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
83C22C_2^2 1−130T2+p2T4 1 - 130 T^{2} + p^{2} T^{4}
89C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
97C2C_2 (1−pT2)2 ( 1 - p 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

−11.67742979012573206868120495160, −11.43657920178970095993615599365, −10.90079055281521496391493721133, −10.60843076004122407086763511364, −10.09034929326524269194283794390, −9.093025499803326964387221212222, −9.070456123070665916918025519974, −8.851926842741046661304085753495, −8.191062176165377562891932796109, −7.80979920227402879121197521368, −6.92755574353117200056568174982, −6.59823322750470748819114060489, −6.39192484901194424268575474235, −5.31681838809082653166237421121, −4.53540613555075457732054973802, −4.39367577433226799856671426843, −3.38471134054077733832054954836, −2.97713835290684456108862309652, −2.18071165988872254726303044855, −1.23341732032264811311135370976, 1.23341732032264811311135370976, 2.18071165988872254726303044855, 2.97713835290684456108862309652, 3.38471134054077733832054954836, 4.39367577433226799856671426843, 4.53540613555075457732054973802, 5.31681838809082653166237421121, 6.39192484901194424268575474235, 6.59823322750470748819114060489, 6.92755574353117200056568174982, 7.80979920227402879121197521368, 8.191062176165377562891932796109, 8.851926842741046661304085753495, 9.070456123070665916918025519974, 9.093025499803326964387221212222, 10.09034929326524269194283794390, 10.60843076004122407086763511364, 10.90079055281521496391493721133, 11.43657920178970095993615599365, 11.67742979012573206868120495160

Graph of the ZZ-function along the critical line