↑

Properties

Label 4-43904-1.1-c1e2-0-13
Degree 44
Conductor 4390443904
Sign −1-1
Analytic cond. 2.799352.79935
Root an. cond. 1.293491.29349
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 7-s − 8-s − 5·9-s − 14-s + 16-s − 9·17-s + 5·18-s − 3·23-s + 2·25-s + 28-s + 4·31-s − 32-s + 9·34-s − 5·36-s − 9·41-s + 3·46-s − 9·47-s + 49-s − 2·50-s − 56-s − 4·62-s − 5·63-s + 64-s − 9·68-s − 15·71-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 5/3·9-s − 0.267·14-s + 1/4·16-s − 2.18·17-s + 1.17·18-s − 0.625·23-s + 2/5·25-s + 0.188·28-s + 0.718·31-s − 0.176·32-s + 1.54·34-s − 5/6·36-s − 1.40·41-s + 0.442·46-s − 1.31·47-s + 1/7·49-s − 0.282·50-s − 0.133·56-s − 0.508·62-s − 0.629·63-s + 1/8·64-s − 1.09·68-s − 1.78·71-s + ⋯

Functional equation

Λ(s)=(43904s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(43904s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 4390443904    =    27⋅732^{7} \cdot 7^{3}
Sign: −1-1
Analytic conductor: 2.799352.79935
Root analytic conductor: 1.293491.29349
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 43904, ( :1/2,1/2), −1)(4,\ 43904,\ (\ :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
7C1C_1 1−T 1 - T
good3C2C_2 (1−T+pT2)(1+T+pT2) ( 1 - T + p T^{2} )( 1 + T + p T^{2} )
5C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
11C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
13C22C_2^2 1−20T2+p2T4 1 - 20 T^{2} + p^{2} T^{4}
17C2C_2×\timesC2C_2 (1+3T+pT2)(1+6T+pT2) ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} )
19C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
23C2C_2×\timesC2C_2 (1−3T+pT2)(1+6T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} )
29C22C_2^2 1+43T2+p2T4 1 + 43 T^{2} + p^{2} T^{4}
31C2C_2×\timesC2C_2 (1−5T+pT2)(1+T+pT2) ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} )
37C22C_2^2 1−5T2+p2T4 1 - 5 T^{2} + p^{2} T^{4}
41C2C_2×\timesC2C_2 (1+pT2)(1+9T+pT2) ( 1 + p T^{2} )( 1 + 9 T + p T^{2} )
43C22C_2^2 1+34T2+p2T4 1 + 34 T^{2} + p^{2} T^{4}
47C2C_2×\timesC2C_2 (1−3T+pT2)(1+12T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 12 T + p T^{2} )
53C22C_2^2 1+85T2+p2T4 1 + 85 T^{2} + p^{2} T^{4}
59C22C_2^2 1−23T2+p2T4 1 - 23 T^{2} + p^{2} T^{4}
61C22C_2^2 1+40T2+p2T4 1 + 40 T^{2} + p^{2} T^{4}
67C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
71C2C_2×\timesC2C_2 (1+6T+pT2)(1+9T+pT2) ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} )
73C2C_2×\timesC2C_2 (1+4T+pT2)(1+10T+pT2) ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} )
79C2C_2×\timesC2C_2 (1+T+pT2)(1+10T+pT2) ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} )
83C22C_2^2 1−5T2+p2T4 1 - 5 T^{2} + p^{2} T^{4}
89C2C_2×\timesC2C_2 (1−12T+pT2)(1+3T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 3 T + p T^{2} )
97C2C_2×\timesC2C_2 (1−14T+pT2)(1−2T+pT2) ( 1 - 14 T + p T^{2} )( 1 - 2 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

−10.16503550298585820097002192537, −9.183105464166945810672815012539, −8.803492526446667316711688403013, −8.565983171959853283933569857530, −8.063621987925375588879513174290, −7.39421188716369306557033815179, −6.70885443291402377233247170593, −6.26828606774537762458311499295, −5.73771297527460382349868991289, −4.92023477868249713444324566679, −4.39611675489762487688262311461, −3.31155823054676521760797309766, −2.63959424907761678348027679278, −1.83998833486204604502597426161, 0, 1.83998833486204604502597426161, 2.63959424907761678348027679278, 3.31155823054676521760797309766, 4.39611675489762487688262311461, 4.92023477868249713444324566679, 5.73771297527460382349868991289, 6.26828606774537762458311499295, 6.70885443291402377233247170593, 7.39421188716369306557033815179, 8.063621987925375588879513174290, 8.565983171959853283933569857530, 8.803492526446667316711688403013, 9.183105464166945810672815012539, 10.16503550298585820097002192537

Graph of the ZZ-function along the critical line