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Properties

Label 4-332928-1.1-c1e2-0-24
Degree 44
Conductor 332928332928
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 2·5-s − 8-s − 3·9-s − 2·10-s − 4·11-s + 16-s + 2·17-s + 3·18-s + 2·20-s + 4·22-s − 6·25-s + 10·29-s − 32-s − 2·34-s − 3·36-s − 10·37-s − 2·40-s − 4·44-s − 6·45-s + 4·47-s + 6·49-s + 6·50-s − 8·55-s − 10·58-s − 2·61-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.353·8-s − 9-s − 0.632·10-s − 1.20·11-s + 1/4·16-s + 0.485·17-s + 0.707·18-s + 0.447·20-s + 0.852·22-s − 6/5·25-s + 1.85·29-s − 0.176·32-s − 0.342·34-s − 1/2·36-s − 1.64·37-s − 0.316·40-s − 0.603·44-s − 0.894·45-s + 0.583·47-s + 6/7·49-s + 0.848·50-s − 1.07·55-s − 1.31·58-s − 0.256·61-s + ⋯

Functional equation

Λ(s)=(332928s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(332928s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\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    =    27⋅32⋅1722^{7} \cdot 3^{2} \cdot 17^{2}
Sign: −1-1
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: 11
Selberg data: (4, 332928, ( :1/2,1/2), −1)(4,\ 332928,\ (\ :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)
bad2C1C_1 1+T 1 + T
3C2C_2 1+pT2 1 + p T^{2}
17C2C_2 1−2T+pT2 1 - 2 T + p T^{2}
good5C2C_2×\timesC2C_2 (1−2T+pT2)(1+pT2) ( 1 - 2 T + p T^{2} )( 1 + p T^{2} )
7C22C_2^2 1−6T2+p2T4 1 - 6 T^{2} + p^{2} T^{4}
11C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
13C22C_2^2 1+14T2+p2T4 1 + 14 T^{2} + p^{2} T^{4}
19C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
23C22C_2^2 1−30T2+p2T4 1 - 30 T^{2} + p^{2} T^{4}
29C2C_2×\timesC2C_2 (1−8T+pT2)(1−2T+pT2) ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} )
31C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
37C2C_2×\timesC2C_2 (1+2T+pT2)(1+8T+pT2) ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} )
41C22C_2^2 1−34T2+p2T4 1 - 34 T^{2} + p^{2} T^{4}
43C22C_2^2 1+38T2+p2T4 1 + 38 T^{2} + p^{2} T^{4}
47C2C_2×\timesC2C_2 (1−4T+pT2)(1+pT2) ( 1 - 4 T + p T^{2} )( 1 + p T^{2} )
53C22C_2^2 1+86T2+p2T4 1 + 86 T^{2} + p^{2} T^{4}
59C22C_2^2 1−110T2+p2T4 1 - 110 T^{2} + p^{2} T^{4}
61C2C_2×\timesC2C_2 (1−8T+pT2)(1+10T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} )
67C22C_2^2 1−46T2+p2T4 1 - 46 T^{2} + p^{2} T^{4}
71C22C_2^2 1+82T2+p2T4 1 + 82 T^{2} + p^{2} T^{4}
73C22C_2^2 1+94T2+p2T4 1 + 94 T^{2} + p^{2} T^{4}
79C22C_2^2 1+10T2+p2T4 1 + 10 T^{2} + p^{2} T^{4}
83C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
89C2C_2×\timesC2C_2 (1+2T+pT2)(1+10T+pT2) ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} )
97C22C_2^2 1+34T2+p2T4 1 + 34 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

−8.532816213622114096654551281017, −8.169314958115618812595298428598, −7.73559972440307903163711231802, −7.21711250154337967288393541385, −6.68155059850002947589758457202, −6.05940964516922099697910222066, −5.79905073177474180187832103866, −5.26787315094652639960258356084, −4.88544323044523213778601090587, −3.94261860012810658825455622412, −3.25182166272888509607356036238, −2.58739387145482707976838456612, −2.25694907809682576965960697336, −1.28168946633399086984930364789, 0, 1.28168946633399086984930364789, 2.25694907809682576965960697336, 2.58739387145482707976838456612, 3.25182166272888509607356036238, 3.94261860012810658825455622412, 4.88544323044523213778601090587, 5.26787315094652639960258356084, 5.79905073177474180187832103866, 6.05940964516922099697910222066, 6.68155059850002947589758457202, 7.21711250154337967288393541385, 7.73559972440307903163711231802, 8.169314958115618812595298428598, 8.532816213622114096654551281017

Graph of the ZZ-function along the critical line