↑

Properties

Label 4-4214-1.1-c1e2-0-0
Degree 44
Conductor 42144214
Sign 11
Analytic cond. 0.2686880.268688
Root an. cond. 0.7199660.719966
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 − 4-s − 7-s + 8-s + 4·9-s + 6·11-s + 14-s + 3·16-s − 4·18-s − 6·22-s − 9·23-s − 25-s + 28-s − 3·32-s − 4·36-s − 14·37-s + 9·43-s − 6·44-s + 9·46-s − 6·49-s + 50-s − 12·53-s − 56-s − 4·63-s − 5·64-s + 10·67-s + 21·71-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 0.377·7-s + 0.353·8-s + 4/3·9-s + 1.80·11-s + 0.267·14-s + 3/4·16-s − 0.942·18-s − 1.27·22-s − 1.87·23-s − 1/5·25-s + 0.188·28-s − 0.530·32-s − 2/3·36-s − 2.30·37-s + 1.37·43-s − 0.904·44-s + 1.32·46-s − 6/7·49-s + 0.141·50-s − 1.64·53-s − 0.133·56-s − 0.503·63-s − 5/8·64-s + 1.22·67-s + 2.49·71-s + ⋯

Functional equation

Λ(s)=(4214s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4214 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4214s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4214 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 42144214    =    2⋅72⋅432 \cdot 7^{2} \cdot 43
Sign: 11
Analytic conductor: 0.2686880.268688
Root analytic conductor: 0.7199660.719966
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 4214, ( :1/2,1/2), 1)(4,\ 4214,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.54127416310.5412741631
L(12)L(\frac12) ≈\approx 0.54127416310.5412741631
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×\timesC2C_2 (1+T)(1+pT2) ( 1 + T )( 1 + p T^{2} )
7C2C_2 1+T+pT2 1 + T + p T^{2}
43C1C_1×\timesC2C_2 (1−T)(1−8T+pT2) ( 1 - T )( 1 - 8 T + p T^{2} )
good3C22C_2^2 1−4T2+p2T4 1 - 4 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} )
11C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
13C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
17C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
19C22C_2^2 1−20T2+p2T4 1 - 20 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} )
29C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
31C22C_2^2 1−26T2+p2T4 1 - 26 T^{2} + p^{2} T^{4}
37C2C_2 (1+7T+pT2)2 ( 1 + 7 T + p T^{2} )^{2}
41C22C_2^2 1+37T2+p2T4 1 + 37 T^{2} + p^{2} T^{4}
47C22C_2^2 1−86T2+p2T4 1 - 86 T^{2} + p^{2} T^{4}
53C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
59C22C_2^2 1+55T2+p2T4 1 + 55 T^{2} + p^{2} T^{4}
61C22C_2^2 1+13T2+p2T4 1 + 13 T^{2} + p^{2} T^{4}
67C2C_2×\timesC2C_2 (1−14T+pT2)(1+4T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} )
71C2C_2×\timesC2C_2 (1−12T+pT2)(1−9T+pT2) ( 1 - 12 T + p T^{2} )( 1 - 9 T + p T^{2} )
73C22C_2^2 1−20T2+p2T4 1 - 20 T^{2} + p^{2} T^{4}
79C2C_2×\timesC2C_2 (1−8T+pT2)(1+10T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} )
83C22C_2^2 1+49T2+p2T4 1 + 49 T^{2} + p^{2} T^{4}
89C22C_2^2 1−128T2+p2T4 1 - 128 T^{2} + p^{2} T^{4}
97C22C_2^2 1−41T2+p2T4 1 - 41 T^{2} + p^{2} T^{4}
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

−12.40992743119000219830426930703, −12.02493042028340985141242906119, −11.19494436916396067256088003266, −10.41052341605502377252150880437, −9.844829833303857857340526608476, −9.505931410169624923239388157236, −8.994621533683078515979595306289, −8.227068586557118849738869505629, −7.62445480258900598893734797038, −6.71059934653250726444651732388, −6.31644324564184210493440834356, −5.19765492894545924172017700826, −4.07724382311546908680728359391, −3.71331134171430874660508363364, −1.61452194687764831160312520057, 1.61452194687764831160312520057, 3.71331134171430874660508363364, 4.07724382311546908680728359391, 5.19765492894545924172017700826, 6.31644324564184210493440834356, 6.71059934653250726444651732388, 7.62445480258900598893734797038, 8.227068586557118849738869505629, 8.994621533683078515979595306289, 9.505931410169624923239388157236, 9.844829833303857857340526608476, 10.41052341605502377252150880437, 11.19494436916396067256088003266, 12.02493042028340985141242906119, 12.40992743119000219830426930703

Graph of the ZZ-function along the critical line