↑

Properties

Label 2-600-1.1-c3-0-12
Degree 22
Conductor 600600
Sign 11
Analytic cond. 35.401135.4011
Root an. cond. 5.949885.94988
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 4·7-s + 9·9-s + 72·11-s + 6·13-s − 38·17-s + 52·19-s − 12·21-s − 152·23-s + 27·27-s − 78·29-s + 120·31-s + 216·33-s + 150·37-s + 18·39-s + 362·41-s + 484·43-s − 280·47-s − 327·49-s − 114·51-s + 670·53-s + 156·57-s + 696·59-s + 222·61-s − 36·63-s + 4·67-s − 456·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.215·7-s + 1/3·9-s + 1.97·11-s + 0.128·13-s − 0.542·17-s + 0.627·19-s − 0.124·21-s − 1.37·23-s + 0.192·27-s − 0.499·29-s + 0.695·31-s + 1.13·33-s + 0.666·37-s + 0.0739·39-s + 1.37·41-s + 1.71·43-s − 0.868·47-s − 0.953·49-s − 0.313·51-s + 1.73·53-s + 0.362·57-s + 1.53·59-s + 0.465·61-s − 0.0719·63-s + 0.00729·67-s − 0.795·69-s + ⋯

Functional equation

Λ(s)=(600s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(600s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 600600    =    23⋅3⋅522^{3} \cdot 3 \cdot 5^{2}
Sign: 11
Analytic conductor: 35.401135.4011
Root analytic conductor: 5.949885.94988
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 600, ( :3/2), 1)(2,\ 600,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 2.7539439962.753943996
L(12)L(\frac12) ≈\approx 2.7539439962.753943996
L(52)L(\frac{5}{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)
bad2 1 1
3 1−pT 1 - p T
5 1 1
good7 1+4T+p3T2 1 + 4 T + p^{3} T^{2}
11 1−72T+p3T2 1 - 72 T + p^{3} T^{2}
13 1−6T+p3T2 1 - 6 T + p^{3} T^{2}
17 1+38T+p3T2 1 + 38 T + p^{3} T^{2}
19 1−52T+p3T2 1 - 52 T + p^{3} T^{2}
23 1+152T+p3T2 1 + 152 T + p^{3} T^{2}
29 1+78T+p3T2 1 + 78 T + p^{3} T^{2}
31 1−120T+p3T2 1 - 120 T + p^{3} T^{2}
37 1−150T+p3T2 1 - 150 T + p^{3} T^{2}
41 1−362T+p3T2 1 - 362 T + p^{3} T^{2}
43 1−484T+p3T2 1 - 484 T + p^{3} T^{2}
47 1+280T+p3T2 1 + 280 T + p^{3} T^{2}
53 1−670T+p3T2 1 - 670 T + p^{3} T^{2}
59 1−696T+p3T2 1 - 696 T + p^{3} T^{2}
61 1−222T+p3T2 1 - 222 T + p^{3} T^{2}
67 1−4T+p3T2 1 - 4 T + p^{3} T^{2}
71 1−96T+p3T2 1 - 96 T + p^{3} T^{2}
73 1+178T+p3T2 1 + 178 T + p^{3} T^{2}
79 1+8pT+p3T2 1 + 8 p T + p^{3} T^{2}
83 1−612T+p3T2 1 - 612 T + p^{3} T^{2}
89 1−994T+p3T2 1 - 994 T + p^{3} T^{2}
97 1+1634T+p3T2 1 + 1634 T + p^{3} T^{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

−9.993368208456742317363106162000, −9.361478696657337687004098864273, −8.652121092913989539432789289794, −7.62675244322544171124379873874, −6.67060950131857582569174570047, −5.88546809392649643222817445964, −4.31275554917054267971754068209, −3.69361082279938996315911108041, −2.30518914621037063047300541490, −1.02510836334262455767194778898, 1.02510836334262455767194778898, 2.30518914621037063047300541490, 3.69361082279938996315911108041, 4.31275554917054267971754068209, 5.88546809392649643222817445964, 6.67060950131857582569174570047, 7.62675244322544171124379873874, 8.652121092913989539432789289794, 9.361478696657337687004098864273, 9.993368208456742317363106162000

Graph of the ZZ-function along the critical line