↑

Properties

Label 2-5733-1.1-c1-0-201
Degree 22
Conductor 57335733
Sign −1-1
Analytic cond. 45.778245.7782
Root an. cond. 6.765966.76596
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.44·2-s + 3.99·4-s − 0.910·5-s + 4.87·8-s − 2.22·10-s − 3.67·11-s + 13-s + 3.95·16-s − 7.18·17-s − 1.97·19-s − 3.63·20-s − 9.00·22-s + 0.596·23-s − 4.17·25-s + 2.44·26-s + 3.64·29-s − 7.08·31-s − 0.0786·32-s − 17.5·34-s + 0.710·37-s − 4.84·38-s − 4.43·40-s − 5.27·41-s + 11.0·43-s − 14.6·44-s + 1.46·46-s − 12.1·47-s + ⋯
L(s)  = 1  + 1.73·2-s + 1.99·4-s − 0.407·5-s + 1.72·8-s − 0.704·10-s − 1.10·11-s + 0.277·13-s + 0.987·16-s − 1.74·17-s − 0.453·19-s − 0.812·20-s − 1.91·22-s + 0.124·23-s − 0.834·25-s + 0.480·26-s + 0.677·29-s − 1.27·31-s − 0.0139·32-s − 3.01·34-s + 0.116·37-s − 0.785·38-s − 0.701·40-s − 0.823·41-s + 1.68·43-s − 2.21·44-s + 0.215·46-s − 1.76·47-s + ⋯

Functional equation

Λ(s)=(5733s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(5733s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 57335733    =    32⋅72⋅133^{2} \cdot 7^{2} \cdot 13
Sign: −1-1
Analytic conductor: 45.778245.7782
Root analytic conductor: 6.765966.76596
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 5733, ( :1/2), −1)(2,\ 5733,\ (\ :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}
ppFp(T)F_p(T)
bad3 1 1
7 1 1
13 1−T 1 - T
good2 1−2.44T+2T2 1 - 2.44T + 2T^{2}
5 1+0.910T+5T2 1 + 0.910T + 5T^{2}
11 1+3.67T+11T2 1 + 3.67T + 11T^{2}
17 1+7.18T+17T2 1 + 7.18T + 17T^{2}
19 1+1.97T+19T2 1 + 1.97T + 19T^{2}
23 1−0.596T+23T2 1 - 0.596T + 23T^{2}
29 1−3.64T+29T2 1 - 3.64T + 29T^{2}
31 1+7.08T+31T2 1 + 7.08T + 31T^{2}
37 1−0.710T+37T2 1 - 0.710T + 37T^{2}
41 1+5.27T+41T2 1 + 5.27T + 41T^{2}
43 1−11.0T+43T2 1 - 11.0T + 43T^{2}
47 1+12.1T+47T2 1 + 12.1T + 47T^{2}
53 1−11.4T+53T2 1 - 11.4T + 53T^{2}
59 1+9.58T+59T2 1 + 9.58T + 59T^{2}
61 1−6.98T+61T2 1 - 6.98T + 61T^{2}
67 1−1.22T+67T2 1 - 1.22T + 67T^{2}
71 1+11.3T+71T2 1 + 11.3T + 71T^{2}
73 1−6.53T+73T2 1 - 6.53T + 73T^{2}
79 1+11.5T+79T2 1 + 11.5T + 79T^{2}
83 1+7.16T+83T2 1 + 7.16T + 83T^{2}
89 1+12.8T+89T2 1 + 12.8T + 89T^{2}
97 1−9.09T+97T2 1 - 9.09T + 97T^{2}
show more
show less
   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−7.43641051499140997212587974607, −6.89908146621326582435040019995, −6.10869383766995218639439079749, −5.52551420064136491400203228167, −4.68867824260123531674919260920, −4.23646699986043749923897263231, −3.42872481169032329473625400646, −2.58534236542313206521991202055, −1.91205652640730474214951925095, 0, 1.91205652640730474214951925095, 2.58534236542313206521991202055, 3.42872481169032329473625400646, 4.23646699986043749923897263231, 4.68867824260123531674919260920, 5.52551420064136491400203228167, 6.10869383766995218639439079749, 6.89908146621326582435040019995, 7.43641051499140997212587974607

Graph of the ZZ-function along the critical line