Properties

Label 4-332928-1.1-c1e2-0-13
Degree 44
Conductor 332928332928
Sign 11
Analytic cond. 21.227721.2277
Root an. cond. 2.146472.14647
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  + 2-s + 4-s + 8-s + 9-s + 16-s + 2·17-s + 18-s − 6·25-s + 32-s + 2·34-s + 36-s + 16·47-s + 10·49-s − 6·50-s + 64-s + 2·68-s + 72-s + 81-s + 20·89-s + 16·94-s + 10·98-s − 6·100-s + 32·103-s − 22·121-s + 127-s + 128-s + 131-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s + 1/3·9-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 6/5·25-s + 0.176·32-s + 0.342·34-s + 1/6·36-s + 2.33·47-s + 10/7·49-s − 0.848·50-s + 1/8·64-s + 0.242·68-s + 0.117·72-s + 1/9·81-s + 2.11·89-s + 1.65·94-s + 1.01·98-s − 3/5·100-s + 3.15·103-s − 2·121-s + 0.0887·127-s + 0.0883·128-s + 0.0873·131-s + ⋯

Functional equation

Λ(s)=(332928s/2ΓC(s)2L(s)=(Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(332928s/2ΓC(s+1/2)2L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 332928332928    =    27321722^{7} \cdot 3^{2} \cdot 17^{2}
Sign: 11
Analytic conductor: 21.227721.2277
Root analytic conductor: 2.146472.14647
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 332928, ( :1/2,1/2), 1)(4,\ 332928,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) \approx 2.9905227352.990522735
L(12)L(\frac12) \approx 2.9905227352.990522735
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2C1C_1 1T 1 - T
3C1C_1×\timesC1C_1 (1T)(1+T) ( 1 - T )( 1 + T )
17C2C_2 12T+pT2 1 - 2 T + p T^{2}
good5C2C_2 (12T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
7C22C_2^2 110T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
11C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
13C2C_2 (16T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
19C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
23C22C_2^2 110T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
29C2C_2 (16T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
31C22C_2^2 1+38T2+p2T4 1 + 38 T^{2} + p^{2} T^{4}
37C2C_2 (12T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
41C2C_2 (1pT2)2 ( 1 - p T^{2} )^{2}
43C2C_2 (14T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
47C2C_2 (18T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
53C2C_2 (16T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
59C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
61C2C_2 (110T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
67C2C_2 (18T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
71C22C_2^2 142T2+p2T4 1 - 42 T^{2} + p^{2} T^{4}
73C2C_2 (16T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
79C22C_2^2 1122T2+p2T4 1 - 122 T^{2} + p^{2} T^{4}
83C2C_2 (116T+pT2)(1+16T+pT2) ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} )
89C2C_2 (110T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
97C22C_2^2 150T2+p2T4 1 - 50 T^{2} + p^{2} T^{4}
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   L(s)=p j=14(1αj,pps)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

−8.839195003001577747571922628762, −8.199397133263483419221021888103, −7.59266086318036482333944944175, −7.47836930577860038386915655621, −6.90369365410745389037013744241, −6.25353947028937099177490903072, −5.90482930646410958798074482335, −5.42848051451252561002388649254, −4.90421879599021250675981877716, −4.23628918591762555005026706755, −3.87002081775942335033180550859, −3.30466060201462864038082665355, −2.49022410853351334524631782583, −1.97065925433832048403434650419, −0.918455563284789218005293309445, 0.918455563284789218005293309445, 1.97065925433832048403434650419, 2.49022410853351334524631782583, 3.30466060201462864038082665355, 3.87002081775942335033180550859, 4.23628918591762555005026706755, 4.90421879599021250675981877716, 5.42848051451252561002388649254, 5.90482930646410958798074482335, 6.25353947028937099177490903072, 6.90369365410745389037013744241, 7.47836930577860038386915655621, 7.59266086318036482333944944175, 8.199397133263483419221021888103, 8.839195003001577747571922628762

Graph of the ZZ-function along the critical line