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Properties

Label 4-332928-1.1-c1e2-0-16
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 no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2·3-s + 4-s + 2·6-s + 8-s + 3·9-s − 8·11-s + 2·12-s + 16-s + 2·17-s + 3·18-s + 8·19-s − 8·22-s + 2·24-s − 6·25-s + 4·27-s + 32-s − 16·33-s + 2·34-s + 3·36-s + 8·38-s + 20·41-s + 24·43-s − 8·44-s + 2·48-s − 14·49-s − 6·50-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s + 0.353·8-s + 9-s − 2.41·11-s + 0.577·12-s + 1/4·16-s + 0.485·17-s + 0.707·18-s + 1.83·19-s − 1.70·22-s + 0.408·24-s − 6/5·25-s + 0.769·27-s + 0.176·32-s − 2.78·33-s + 0.342·34-s + 1/2·36-s + 1.29·38-s + 3.12·41-s + 3.65·43-s − 1.20·44-s + 0.288·48-s − 2·49-s − 0.848·50-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: 11
Analytic conductor: 21.227721.2277
Root analytic conductor: 2.146472.14647
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
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 4.1578817144.157881714
L(12)L(\frac12) ≈\approx 4.1578817144.157881714
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
3C1C_1 (1−T)2 ( 1 - T )^{2}
17C1C_1 (1−T)2 ( 1 - T )^{2}
good5C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
7C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
11C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
13C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
19C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
29C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
31C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
37C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
41C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
43C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
47C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
53C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
59C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
61C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
67C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
79C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
83C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
89C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
97C2C_2 (1+14T+pT2)2 ( 1 + 14 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

−8.658113684153939465660001552715, −7.958882066985845549410362974675, −7.890763986910657579747142309970, −7.36440720073199929079936076774, −7.34292617858622580122915984824, −6.29236664701447308952901966222, −5.62906231402718523510730620943, −5.51457882926074302977186886250, −4.91746541938042376263088347369, −4.13875090005906653574798019221, −3.84610425166553964062381179508, −2.86767294411575254972359726206, −2.76309813409348720224135008444, −2.21434037861733189800471556077, −1.01738437769258744176787321955, 1.01738437769258744176787321955, 2.21434037861733189800471556077, 2.76309813409348720224135008444, 2.86767294411575254972359726206, 3.84610425166553964062381179508, 4.13875090005906653574798019221, 4.91746541938042376263088347369, 5.51457882926074302977186886250, 5.62906231402718523510730620943, 6.29236664701447308952901966222, 7.34292617858622580122915984824, 7.36440720073199929079936076774, 7.890763986910657579747142309970, 7.958882066985845549410362974675, 8.658113684153939465660001552715

Graph of the ZZ-function along the critical line