↑

Properties

Label 4-1280e2-1.1-c1e2-0-50
Degree 44
Conductor 16384001638400
Sign 11
Analytic cond. 104.465104.465
Root an. cond. 3.197003.19700
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4·7-s + 2·9-s + 4·17-s + 12·23-s − 25-s + 8·31-s + 20·41-s − 4·47-s − 2·49-s + 8·63-s + 24·71-s − 20·73-s − 16·79-s − 5·81-s + 12·89-s + 20·97-s − 4·103-s − 12·113-s + 16·119-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 8·153-s + ⋯
L(s)  = 1  + 1.51·7-s + 2/3·9-s + 0.970·17-s + 2.50·23-s − 1/5·25-s + 1.43·31-s + 3.12·41-s − 0.583·47-s − 2/7·49-s + 1.00·63-s + 2.84·71-s − 2.34·73-s − 1.80·79-s − 5/9·81-s + 1.27·89-s + 2.03·97-s − 0.394·103-s − 1.12·113-s + 1.46·119-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.646·153-s + ⋯

Functional equation

Λ(s)=(1638400s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1638400s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 16384001638400    =    216⋅522^{16} \cdot 5^{2}
Sign: 11
Analytic conductor: 104.465104.465
Root analytic conductor: 3.197003.19700
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 1638400, ( :1/2,1/2), 1)(4,\ 1638400,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.7151369293.715136929
L(12)L(\frac12) ≈\approx 3.7151369293.715136929
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)
bad2 1 1
5C2C_2 1+T2 1 + T^{2}
good3C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
7C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
11C22C_2^2 1−6T2+p2T4 1 - 6 T^{2} + p^{2} T^{4}
13C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
17C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
19C22C_2^2 1+26T2+p2T4 1 + 26 T^{2} + p^{2} T^{4}
23C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
29C22C_2^2 1−54T2+p2T4 1 - 54 T^{2} + p^{2} T^{4}
31C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
37C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
41C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
43C22C_2^2 1−82T2+p2T4 1 - 82 T^{2} + p^{2} T^{4}
47C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
53C22C_2^2 1−102T2+p2T4 1 - 102 T^{2} + p^{2} T^{4}
59C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
61C22C_2^2 1−118T2+p2T4 1 - 118 T^{2} + p^{2} T^{4}
67C22C_2^2 1−98T2+p2T4 1 - 98 T^{2} + p^{2} T^{4}
71C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
73C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
79C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
83C22C_2^2 1−66T2+p2T4 1 - 66 T^{2} + p^{2} T^{4}
89C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
97C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{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

−9.776200963334991338265089865452, −9.548542620219817254205937044250, −9.046866256524912021517874461995, −8.513715621080274122741456904179, −8.378850205366672997921174603243, −7.68613461182798119360937015743, −7.42789027444500383537495659354, −7.33199107908077772433713637004, −6.41557191448445407871384308461, −6.31521360589688984426753952956, −5.54024007330661594377694783736, −5.17736275527103570162766340915, −4.71579075765337062072571809757, −4.48420504069430678262865452901, −3.89817626275689786006102587650, −3.17743772980011364424921909151, −2.74380996083893799449863223262, −2.07445675364317249813953668184, −1.14278176642425696447911837790, −1.08571203973696756141475815507, 1.08571203973696756141475815507, 1.14278176642425696447911837790, 2.07445675364317249813953668184, 2.74380996083893799449863223262, 3.17743772980011364424921909151, 3.89817626275689786006102587650, 4.48420504069430678262865452901, 4.71579075765337062072571809757, 5.17736275527103570162766340915, 5.54024007330661594377694783736, 6.31521360589688984426753952956, 6.41557191448445407871384308461, 7.33199107908077772433713637004, 7.42789027444500383537495659354, 7.68613461182798119360937015743, 8.378850205366672997921174603243, 8.513715621080274122741456904179, 9.046866256524912021517874461995, 9.548542620219817254205937044250, 9.776200963334991338265089865452

Graph of the ZZ-function along the critical line