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Properties

Label 4-24448-1.1-c1e2-0-2
Degree 44
Conductor 2444824448
Sign 11
Analytic cond. 1.558821.55882
Root an. cond. 1.117371.11737
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 + 2·7-s + 8-s + 2·9-s + 2·14-s + 16-s − 17-s + 2·18-s − 14·23-s − 25-s + 2·28-s + 6·31-s + 32-s − 34-s + 2·36-s − 5·41-s − 14·46-s + 7·47-s + 5·49-s − 50-s + 2·56-s + 6·62-s + 4·63-s + 64-s − 68-s − 26·71-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.755·7-s + 0.353·8-s + 2/3·9-s + 0.534·14-s + 1/4·16-s − 0.242·17-s + 0.471·18-s − 2.91·23-s − 1/5·25-s + 0.377·28-s + 1.07·31-s + 0.176·32-s − 0.171·34-s + 1/3·36-s − 0.780·41-s − 2.06·46-s + 1.02·47-s + 5/7·49-s − 0.141·50-s + 0.267·56-s + 0.762·62-s + 0.503·63-s + 1/8·64-s − 0.121·68-s − 3.08·71-s + ⋯

Functional equation

Λ(s)=(24448s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 24448 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(24448s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 24448 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 2444824448    =    27⋅1912^{7} \cdot 191
Sign: 11
Analytic conductor: 1.558821.55882
Root analytic conductor: 1.117371.11737
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 24448, ( :1/2,1/2), 1)(4,\ 24448,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.9626114641.962611464
L(12)L(\frac12) ≈\approx 1.9626114641.962611464
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
191C1C_1×\timesC2C_2 (1−T)(1−6T+pT2) ( 1 - T )( 1 - 6 T + p T^{2} )
good3C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
5C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
7C2C_2×\timesC2C_2 (1−5T+pT2)(1+3T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 3 T + p T^{2} )
11C22C_2^2 1−5T2+p2T4 1 - 5 T^{2} + p^{2} T^{4}
13C22C_2^2 1−19T2+p2T4 1 - 19 T^{2} + p^{2} T^{4}
17C2C_2×\timesC2C_2 (1−5T+pT2)(1+6T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 6 T + p T^{2} )
19C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
23C2C_2×\timesC2C_2 (1+6T+pT2)(1+8T+pT2) ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} )
29C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
31C2C_2×\timesC2C_2 (1−7T+pT2)(1+T+pT2) ( 1 - 7 T + p T^{2} )( 1 + T + p T^{2} )
37C22C_2^2 1+16T2+p2T4 1 + 16 T^{2} + p^{2} T^{4}
41C2C_2×\timesC2C_2 (1−5T+pT2)(1+10T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 10 T + p T^{2} )
43C22C_2^2 1+8T2+p2T4 1 + 8 T^{2} + p^{2} T^{4}
47C2C_2×\timesC2C_2 (1−7T+pT2)(1+pT2) ( 1 - 7 T + p T^{2} )( 1 + p T^{2} )
53C22C_2^2 1−4T2+p2T4 1 - 4 T^{2} + p^{2} T^{4}
59C22C_2^2 1−8T2+p2T4 1 - 8 T^{2} + p^{2} T^{4}
61C22C_2^2 1−104T2+p2T4 1 - 104 T^{2} + p^{2} T^{4}
67C22C_2^2 1+94T2+p2T4 1 + 94 T^{2} + p^{2} T^{4}
71C2C_2×\timesC2C_2 (1+12T+pT2)(1+14T+pT2) ( 1 + 12 T + p T^{2} )( 1 + 14 T + p T^{2} )
73C2C_2×\timesC2C_2 (1−16T+pT2)(1−T+pT2) ( 1 - 16 T + p T^{2} )( 1 - T + p T^{2} )
79C2C_2×\timesC2C_2 (1−8T+pT2)(1+17T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 17 T + p T^{2} )
83C22C_2^2 1−146T2+p2T4 1 - 146 T^{2} + p^{2} T^{4}
89C2C_2×\timesC2C_2 (1−10T+pT2)(1−T+pT2) ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} )
97C2C_2 (1−5T+pT2)(1+5T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{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

−10.62568836253306071673732054988, −10.17407737261276440003145294717, −9.947135977322009842513373671912, −9.042682854241946217968178648794, −8.425563676112510119340051542023, −7.86765929867489967440726902719, −7.48219687920498682034105288097, −6.72927967112185856729530136877, −6.08297860483442248886341827672, −5.63490496466260502638218294432, −4.72618195601609781932857717580, −4.27399667014652757006015136100, −3.67975111074027991085489251918, −2.48183014565161592351785890256, −1.67329874314093529772851850688, 1.67329874314093529772851850688, 2.48183014565161592351785890256, 3.67975111074027991085489251918, 4.27399667014652757006015136100, 4.72618195601609781932857717580, 5.63490496466260502638218294432, 6.08297860483442248886341827672, 6.72927967112185856729530136877, 7.48219687920498682034105288097, 7.86765929867489967440726902719, 8.425563676112510119340051542023, 9.042682854241946217968178648794, 9.947135977322009842513373671912, 10.17407737261276440003145294717, 10.62568836253306071673732054988

Graph of the ZZ-function along the critical line