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Properties

Label 4-480e2-1.1-c1e2-0-34
Degree 44
Conductor 230400230400
Sign −1-1
Analytic cond. 14.690514.6905
Root an. cond. 1.957751.95775
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 11

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·7-s + 9-s − 4·17-s + 16·23-s + 25-s + 20·41-s − 16·47-s + 34·49-s − 8·63-s − 28·73-s − 32·79-s + 81-s + 4·89-s + 4·97-s − 8·103-s + 12·113-s + 32·119-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 4·153-s + 157-s − 128·161-s + ⋯
L(s)  = 1  − 3.02·7-s + 1/3·9-s − 0.970·17-s + 3.33·23-s + 1/5·25-s + 3.12·41-s − 2.33·47-s + 34/7·49-s − 1.00·63-s − 3.27·73-s − 3.60·79-s + 1/9·81-s + 0.423·89-s + 0.406·97-s − 0.788·103-s + 1.12·113-s + 2.93·119-s − 2·121-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 − 0.323·153-s + 0.0798·157-s − 10.0·161-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: −1-1
Analytic conductor: 14.690514.6905
Root analytic conductor: 1.957751.95775
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 230400, ( :1/2,1/2), −1)(4,\ 230400,\ (\ :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)
bad2 1 1
3C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
5C1C_1×\timesC1C_1 (1−T)(1+T) ( 1 - T )( 1 + T )
good7C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
11C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
13C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
17C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
19C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
23C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
29C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
31C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
37C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
41C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
43C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
47C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
53C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
59C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
61C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
67C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C2C_2 (1+14T+pT2)2 ( 1 + 14 T + p T^{2} )^{2}
79C2C_2 (1+16T+pT2)2 ( 1 + 16 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 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
97C2C_2 (1−2T+pT2)2 ( 1 - 2 T + 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

−8.931256392945798101264819577475, −8.620274137521206455931441584726, −7.45166089807995498253197140252, −7.31616880556813656693675020931, −6.73102949270507316916720435534, −6.49211773175788373093258725474, −6.00871420537581383609527234257, −5.45298154060946963380399619268, −4.63883887090438359185158186412, −4.19366402654330570965762069528, −3.39015414339768891989414504202, −2.85965731186513405981248890470, −2.75290686228331533750926083473, −1.16849564057756810686392475492, 0, 1.16849564057756810686392475492, 2.75290686228331533750926083473, 2.85965731186513405981248890470, 3.39015414339768891989414504202, 4.19366402654330570965762069528, 4.63883887090438359185158186412, 5.45298154060946963380399619268, 6.00871420537581383609527234257, 6.49211773175788373093258725474, 6.73102949270507316916720435534, 7.31616880556813656693675020931, 7.45166089807995498253197140252, 8.620274137521206455931441584726, 8.931256392945798101264819577475

Graph of the ZZ-function along the critical line