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Properties

Label 4-987696-1.1-c1e2-0-7
Degree 44
Conductor 987696987696
Sign −1-1
Analytic cond. 62.976362.9763
Root an. cond. 2.817042.81704
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 6·9-s + 3·13-s + 19-s + 3·25-s − 9·27-s − 12·31-s − 6·37-s − 9·39-s + 5·43-s − 5·49-s − 3·57-s + 17·61-s − 3·67-s + 9·73-s − 9·75-s − 9·79-s + 9·81-s + 36·93-s + 6·97-s − 6·109-s + 18·111-s + 18·117-s + 6·121-s + 127-s − 15·129-s + 131-s + ⋯
L(s)  = 1  − 1.73·3-s + 2·9-s + 0.832·13-s + 0.229·19-s + 3/5·25-s − 1.73·27-s − 2.15·31-s − 0.986·37-s − 1.44·39-s + 0.762·43-s − 5/7·49-s − 0.397·57-s + 2.17·61-s − 0.366·67-s + 1.05·73-s − 1.03·75-s − 1.01·79-s + 81-s + 3.73·93-s + 0.609·97-s − 0.574·109-s + 1.70·111-s + 1.66·117-s + 6/11·121-s + 0.0887·127-s − 1.32·129-s + 0.0873·131-s + ⋯

Functional equation

Λ(s)=(987696s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 987696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(987696s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 987696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 987696987696    =    24⋅32⋅1932^{4} \cdot 3^{2} \cdot 19^{3}
Sign: −1-1
Analytic conductor: 62.976362.9763
Root analytic conductor: 2.817042.81704
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 987696, ( :1/2,1/2), −1)(4,\ 987696,\ (\ :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
3C2C_2 1+pT+pT2 1 + p T + p T^{2}
19C1C_1 1−T 1 - T
good5C22C_2^2 1−3T2+p2T4 1 - 3 T^{2} + p^{2} T^{4}
7C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
11C22C_2^2 1−6T2+p2T4 1 - 6 T^{2} + p^{2} T^{4}
13C2C_2 (1−5T+pT2)(1+2T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} )
17C22C_2^2 1+29T2+p2T4 1 + 29 T^{2} + p^{2} T^{4}
23C2C_2 (1−5T+pT2)(1+5T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} )
29C22C_2^2 1−15T2+p2T4 1 - 15 T^{2} + p^{2} T^{4}
31C2C_2×\timesC2C_2 (1+5T+pT2)(1+7T+pT2) ( 1 + 5 T + p T^{2} )( 1 + 7 T + p T^{2} )
37C2C_2×\timesC2C_2 (1−T+pT2)(1+7T+pT2) ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} )
41C22C_2^2 1+21T2+p2T4 1 + 21 T^{2} + p^{2} T^{4}
43C2C_2×\timesC2C_2 (1−4T+pT2)(1−T+pT2) ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} )
47C22C_2^2 1−63T2+p2T4 1 - 63 T^{2} + p^{2} T^{4}
53C2C_2 (1−11T+pT2)(1+11T+pT2) ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} )
59C22C_2^2 1+T2+p2T4 1 + T^{2} + p^{2} T^{4}
61C2C_2×\timesC2C_2 (1−10T+pT2)(1−7T+pT2) ( 1 - 10 T + p T^{2} )( 1 - 7 T + p T^{2} )
67C2C_2×\timesC2C_2 (1−9T+pT2)(1+12T+pT2) ( 1 - 9 T + p T^{2} )( 1 + 12 T + p T^{2} )
71C22C_2^2 1+57T2+p2T4 1 + 57 T^{2} + p^{2} T^{4}
73C2C_2×\timesC2C_2 (1−15T+pT2)(1+6T+pT2) ( 1 - 15 T + p T^{2} )( 1 + 6 T + p T^{2} )
79C2C_2×\timesC2C_2 (1+T+pT2)(1+8T+pT2) ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} )
83C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
89C2C_2 (1−15T+pT2)(1+15T+pT2) ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} )
97C2C_2×\timesC2C_2 (1−13T+pT2)(1+7T+pT2) ( 1 - 13 T + p T^{2} )( 1 + 7 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

−7.65478542993864032991661837675, −7.36028396690067987832949372847, −6.97962741649462758989451840867, −6.42454560680663576093867154103, −6.20130789958900120032074211076, −5.60043213822687702964922228729, −5.22610279522421829320276692403, −5.02175821113324799428203151522, −4.24507010736908696665397610267, −3.80392351122560825863494767467, −3.36903337610250286320755243958, −2.40344357012843977113173240461, −1.62531368494710710101103096307, −0.999909831714497685598503526662, 0, 0.999909831714497685598503526662, 1.62531368494710710101103096307, 2.40344357012843977113173240461, 3.36903337610250286320755243958, 3.80392351122560825863494767467, 4.24507010736908696665397610267, 5.02175821113324799428203151522, 5.22610279522421829320276692403, 5.60043213822687702964922228729, 6.20130789958900120032074211076, 6.42454560680663576093867154103, 6.97962741649462758989451840867, 7.36028396690067987832949372847, 7.65478542993864032991661837675

Graph of the ZZ-function along the critical line