↑

Properties

Label 4-332928-1.1-c1e2-0-3
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2·3-s + 4-s − 2·6-s + 8-s + 3·9-s − 2·12-s + 16-s + 2·17-s + 3·18-s − 2·24-s − 6·25-s − 4·27-s + 32-s + 2·34-s + 3·36-s − 12·41-s + 8·43-s − 2·48-s + 10·49-s − 6·50-s − 4·51-s − 4·54-s + 16·59-s + 64-s + 2·68-s + 3·72-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 − 0.577·12-s + 1/4·16-s + 0.485·17-s + 0.707·18-s − 0.408·24-s − 6/5·25-s − 0.769·27-s + 0.176·32-s + 0.342·34-s + 1/2·36-s − 1.87·41-s + 1.21·43-s − 0.288·48-s + 10/7·49-s − 0.848·50-s − 0.560·51-s − 0.544·54-s + 2.08·59-s + 1/8·64-s + 0.242·68-s + 0.353·72-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: 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 1.8007871281.800787128
L(12)L(\frac12) ≈\approx 1.8007871281.800787128
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} )
7C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
11C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
13C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
19C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
23C22C_2^2 1−26T2+p2T4 1 - 26 T^{2} + p^{2} T^{4}
29C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
31C22C_2^2 1−26T2+p2T4 1 - 26 T^{2} + p^{2} T^{4}
37C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
41C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
43C2C_2×\timesC2C_2 (1−12T+pT2)(1+4T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} )
47C22C_2^2 1+62T2+p2T4 1 + 62 T^{2} + p^{2} T^{4}
53C22C_2^2 1−74T2+p2T4 1 - 74 T^{2} + p^{2} T^{4}
59C2C_2×\timesC2C_2 (1−12T+pT2)(1−4T+pT2) ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} )
61C22C_2^2 1−58T2+p2T4 1 - 58 T^{2} + p^{2} T^{4}
67C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
71C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
73C2C_2×\timesC2C_2 (1−10T+pT2)(1+6T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} )
79C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
83C2C_2×\timesC2C_2 (1−12T+pT2)(1−4T+pT2) ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} )
89C2C_2×\timesC2C_2 (1−10T+pT2)(1+6T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} )
97C2C_2×\timesC2C_2 (1−10T+pT2)(1+14T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} )
show more
show less
   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

−8.661140335457030953268886751361, −8.234369008581860367630742160110, −7.65049983315333815787742157938, −7.23007339071516806375060687388, −6.78961807332716027516059567573, −6.31545545210484520681532449235, −5.77190626255755447945500332687, −5.46817165791435838365201502987, −5.02683125206524074860344164800, −4.39174104486575572646924771442, −3.87028361929848019243608003645, −3.41714102710223844709096907143, −2.45289756185632805884410446281, −1.79327144070165485772745385168, −0.74719969981905527974315558451, 0.74719969981905527974315558451, 1.79327144070165485772745385168, 2.45289756185632805884410446281, 3.41714102710223844709096907143, 3.87028361929848019243608003645, 4.39174104486575572646924771442, 5.02683125206524074860344164800, 5.46817165791435838365201502987, 5.77190626255755447945500332687, 6.31545545210484520681532449235, 6.78961807332716027516059567573, 7.23007339071516806375060687388, 7.65049983315333815787742157938, 8.234369008581860367630742160110, 8.661140335457030953268886751361

Graph of the ZZ-function along the critical line