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Properties

Label 4-588e2-1.1-c1e2-0-31
Degree 44
Conductor 345744345744
Sign −1-1
Analytic cond. 22.044922.0449
Root an. cond. 2.166842.16684
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  + 2-s − 4-s − 3·8-s − 9-s + 2·11-s − 16-s − 18-s + 2·22-s + 2·23-s − 4·25-s − 8·29-s + 5·32-s + 36-s − 2·44-s + 2·46-s − 4·50-s + 8·53-s − 8·58-s + 7·64-s − 12·67-s − 18·71-s + 3·72-s − 4·79-s + 81-s − 6·88-s − 2·92-s − 2·99-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 1.06·8-s − 1/3·9-s + 0.603·11-s − 1/4·16-s − 0.235·18-s + 0.426·22-s + 0.417·23-s − 4/5·25-s − 1.48·29-s + 0.883·32-s + 1/6·36-s − 0.301·44-s + 0.294·46-s − 0.565·50-s + 1.09·53-s − 1.05·58-s + 7/8·64-s − 1.46·67-s − 2.13·71-s + 0.353·72-s − 0.450·79-s + 1/9·81-s − 0.639·88-s − 0.208·92-s − 0.201·99-s + ⋯

Functional equation

Λ(s)=(345744s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(345744s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 345744345744    =    24⋅32⋅742^{4} \cdot 3^{2} \cdot 7^{4}
Sign: −1-1
Analytic conductor: 22.044922.0449
Root analytic conductor: 2.166842.16684
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 345744, ( :1/2,1/2), −1)(4,\ 345744,\ (\ :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)
bad2C2C_2 1−T+pT2 1 - T + p T^{2}
3C2C_2 1+T2 1 + T^{2}
7 1 1
good5C22C_2^2 1+4T2+p2T4 1 + 4 T^{2} + p^{2} T^{4}
11C2C_2×\timesC2C_2 (1−4T+pT2)(1+2T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} )
13C22C_2^2 1+8T2+p2T4 1 + 8 T^{2} + p^{2} T^{4}
17C22C_2^2 1+20T2+p2T4 1 + 20 T^{2} + p^{2} T^{4}
19C22C_2^2 1+14T2+p2T4 1 + 14 T^{2} + p^{2} T^{4}
23C2C_2×\timesC2C_2 (1−2T+pT2)(1+pT2) ( 1 - 2 T + p T^{2} )( 1 + p T^{2} )
29C2C_2×\timesC2C_2 (1+2T+pT2)(1+6T+pT2) ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} )
31C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
37C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
41C22C_2^2 1−20T2+p2T4 1 - 20 T^{2} + p^{2} T^{4}
43C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
47C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
53C2C_2×\timesC2C_2 (1−14T+pT2)(1+6T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} )
59C22C_2^2 1+14T2+p2T4 1 + 14 T^{2} + p^{2} T^{4}
61C22C_2^2 1+16T2+p2T4 1 + 16 T^{2} + p^{2} T^{4}
67C2C_2×\timesC2C_2 (1−4T+pT2)(1+16T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 16 T + p T^{2} )
71C2C_2×\timesC2C_2 (1+8T+pT2)(1+10T+pT2) ( 1 + 8 T + p T^{2} )( 1 + 10 T + p T^{2} )
73C22C_2^2 1−40T2+p2T4 1 - 40 T^{2} + p^{2} T^{4}
79C2C_2×\timesC2C_2 (1−4T+pT2)(1+8T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} )
83C22C_2^2 1+26T2+p2T4 1 + 26 T^{2} + p^{2} T^{4}
89C22C_2^2 1+156T2+p2T4 1 + 156 T^{2} + p^{2} T^{4}
97C22C_2^2 1+40T2+p2T4 1 + 40 T^{2} + p^{2} T^{4}
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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

−8.739669822998969445809140863235, −8.034516887502443310649335442709, −7.52685283860693278740387088025, −7.15143906922658593640837830289, −6.45585929004488164739050655816, −5.99444197952005469662508385604, −5.64425540196005984867157019856, −5.16050516489234500565730154860, −4.52358344101759891331149131012, −4.06963482182445630773446579280, −3.58887616485273778765261206279, −3.02437764916907353123409236856, −2.28485132977309640466176859544, −1.33358386294238772488720373779, 0, 1.33358386294238772488720373779, 2.28485132977309640466176859544, 3.02437764916907353123409236856, 3.58887616485273778765261206279, 4.06963482182445630773446579280, 4.52358344101759891331149131012, 5.16050516489234500565730154860, 5.64425540196005984867157019856, 5.99444197952005469662508385604, 6.45585929004488164739050655816, 7.15143906922658593640837830289, 7.52685283860693278740387088025, 8.034516887502443310649335442709, 8.739669822998969445809140863235

Graph of the ZZ-function along the critical line