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Properties

Label 4-480e2-1.1-c1e2-0-27
Degree 44
Conductor 230400230400
Sign 11
Analytic cond. 14.690514.6905
Root an. cond. 1.957751.95775
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·7-s + 9-s + 4·23-s − 25-s + 4·41-s + 12·47-s − 2·49-s + 4·63-s + 8·71-s − 4·73-s + 81-s + 4·89-s − 4·97-s + 12·103-s − 8·113-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 16·161-s + 163-s + 167-s + 6·169-s + ⋯
L(s)  = 1  + 1.51·7-s + 1/3·9-s + 0.834·23-s − 1/5·25-s + 0.624·41-s + 1.75·47-s − 2/7·49-s + 0.503·63-s + 0.949·71-s − 0.468·73-s + 1/9·81-s + 0.423·89-s − 0.406·97-s + 1.18·103-s − 0.752·113-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 1.26·161-s + 0.0783·163-s + 0.0773·167-s + 6/13·169-s + ⋯

Functional equation

Λ(s)=(230400s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 230400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(230400s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 230400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 230400230400    =    210⋅32⋅522^{10} \cdot 3^{2} \cdot 5^{2}
Sign: 11
Analytic conductor: 14.690514.6905
Root analytic conductor: 1.957751.95775
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 230400, ( :1/2,1/2), 1)(4,\ 230400,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.2215875582.221587558
L(12)L(\frac12) ≈\approx 2.2215875582.221587558
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×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
5C2C_2 1+T2 1 + T^{2}
good7C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
11C22C_2^2 1+10T2+p2T4 1 + 10 T^{2} + p^{2} T^{4}
13C22C_2^2 1−6T2+p2T4 1 - 6 T^{2} + p^{2} T^{4}
17C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
19C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
23C2C_2×\timesC2C_2 (1−4T+pT2)(1+pT2) ( 1 - 4 T + p T^{2} )( 1 + p T^{2} )
29C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
31C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
37C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
41C2C_2×\timesC2C_2 (1−6T+pT2)(1+2T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} )
43C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
47C2C_2×\timesC2C_2 (1−12T+pT2)(1+pT2) ( 1 - 12 T + p T^{2} )( 1 + p T^{2} )
53C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
59C22C_2^2 1+90T2+p2T4 1 + 90 T^{2} + p^{2} T^{4}
61C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
67C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
71C2C_2×\timesC2C_2 (1−8T+pT2)(1+pT2) ( 1 - 8 T + p T^{2} )( 1 + p T^{2} )
73C2C_2×\timesC2C_2 (1−10T+pT2)(1+14T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} )
79C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
83C22C_2^2 1−58T2+p2T4 1 - 58 T^{2} + p^{2} T^{4}
89C2C_2×\timesC2C_2 (1−6T+pT2)(1+2T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} )
97C2C_2×\timesC2C_2 (1−2T+pT2)(1+6T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 6 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

−9.010491145617605083476351870773, −8.460687659062073508331132456542, −8.061548317513178034733944314162, −7.59291462867186493700473932851, −7.24489141892793086305399814747, −6.65388597794988806319202028461, −6.07248787764964529117609675479, −5.45297028291658394314709952790, −5.03796141753130864842855351444, −4.51416044297849093294124555918, −4.06028195628447761837367673348, −3.30054061241319217567250675863, −2.48949295169962412612476552574, −1.80359649052988157033946617775, −1.00760419767862181709979731337, 1.00760419767862181709979731337, 1.80359649052988157033946617775, 2.48949295169962412612476552574, 3.30054061241319217567250675863, 4.06028195628447761837367673348, 4.51416044297849093294124555918, 5.03796141753130864842855351444, 5.45297028291658394314709952790, 6.07248787764964529117609675479, 6.65388597794988806319202028461, 7.24489141892793086305399814747, 7.59291462867186493700473932851, 8.061548317513178034733944314162, 8.460687659062073508331132456542, 9.010491145617605083476351870773

Graph of the ZZ-function along the critical line