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Properties

Label 4-768e2-1.1-c1e2-0-6
Degree 44
Conductor 589824589824
Sign 11
Analytic cond. 37.607637.6076
Root an. cond. 2.476392.47639
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 + 12·17-s − 8·19-s + 2·25-s − 4·27-s + 12·41-s − 8·43-s − 2·49-s − 24·51-s + 16·57-s + 24·59-s + 8·67-s − 4·73-s − 4·75-s + 5·81-s − 12·89-s − 4·97-s + 24·107-s − 12·113-s − 22·121-s − 24·123-s + 127-s + 16·129-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 1.15·3-s + 9-s + 2.91·17-s − 1.83·19-s + 2/5·25-s − 0.769·27-s + 1.87·41-s − 1.21·43-s − 2/7·49-s − 3.36·51-s + 2.11·57-s + 3.12·59-s + 0.977·67-s − 0.468·73-s − 0.461·75-s + 5/9·81-s − 1.27·89-s − 0.406·97-s + 2.32·107-s − 1.12·113-s − 2·121-s − 2.16·123-s + 0.0887·127-s + 1.40·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

Λ(s)=(589824s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(589824s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 589824589824    =    216⋅322^{16} \cdot 3^{2}
Sign: 11
Analytic conductor: 37.607637.6076
Root analytic conductor: 2.476392.47639
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 589824, ( :1/2,1/2), 1)(4,\ 589824,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.2889202751.288920275
L(12)L(\frac12) ≈\approx 1.2889202751.288920275
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}
good5C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
7C22C_2^2 1+2T2+p2T4 1 + 2 T^{2} + p^{2} T^{4}
11C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
13C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
17C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
19C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
23C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
29C22C_2^2 1+46T2+p2T4 1 + 46 T^{2} + p^{2} T^{4}
31C22C_2^2 1+50T2+p2T4 1 + 50 T^{2} + p^{2} T^{4}
37C22C_2^2 1+26T2+p2T4 1 + 26 T^{2} + p^{2} T^{4}
41C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
43C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
47C22C_2^2 1+46T2+p2T4 1 + 46 T^{2} + p^{2} T^{4}
53C22C_2^2 1+94T2+p2T4 1 + 94 T^{2} + p^{2} T^{4}
59C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
61C22C_2^2 1+74T2+p2T4 1 + 74 T^{2} + p^{2} T^{4}
67C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
71C22C_2^2 1+94T2+p2T4 1 + 94 T^{2} + p^{2} T^{4}
73C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
79C22C_2^2 1+50T2+p2T4 1 + 50 T^{2} + p^{2} T^{4}
83C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
89C2C_2 (1+6T+pT2)2 ( 1 + 6 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

−10.40832323178950496516690073363, −10.24142541407562571412619631750, −9.757942827002881799281795958243, −9.567824670349199995278608367568, −8.600997925327965764229099302844, −8.546702118283194580045683006223, −7.79344604086434859189305136321, −7.60853996407630508849648715281, −6.89673040000006302638164559861, −6.63629104888727364493423594851, −6.01226049225962676935830219852, −5.67974700076737418286177162481, −5.27210138874681810345572871230, −4.82532112152912354925956242158, −4.06798219088688365647526235101, −3.79580701790612401048742900272, −3.02866888416199003373007631606, −2.28313200304748551834209679982, −1.39673240493110064978376001926, −0.68078562470568089929410367530, 0.68078562470568089929410367530, 1.39673240493110064978376001926, 2.28313200304748551834209679982, 3.02866888416199003373007631606, 3.79580701790612401048742900272, 4.06798219088688365647526235101, 4.82532112152912354925956242158, 5.27210138874681810345572871230, 5.67974700076737418286177162481, 6.01226049225962676935830219852, 6.63629104888727364493423594851, 6.89673040000006302638164559861, 7.60853996407630508849648715281, 7.79344604086434859189305136321, 8.546702118283194580045683006223, 8.600997925327965764229099302844, 9.567824670349199995278608367568, 9.757942827002881799281795958243, 10.24142541407562571412619631750, 10.40832323178950496516690073363

Graph of the ZZ-function along the critical line