↑

Properties

Label 2-1680-1.1-c3-0-65
Degree 22
Conductor 16801680
Sign −1-1
Analytic cond. 99.123299.1232
Root an. cond. 9.956069.95606
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 5·5-s − 7·7-s + 9·9-s − 42·11-s + 20·13-s + 15·15-s + 66·17-s − 38·19-s − 21·21-s − 12·23-s + 25·25-s + 27·27-s − 258·29-s − 146·31-s − 126·33-s − 35·35-s + 434·37-s + 60·39-s − 282·41-s − 20·43-s + 45·45-s + 72·47-s + 49·49-s + 198·51-s + 336·53-s − 210·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.15·11-s + 0.426·13-s + 0.258·15-s + 0.941·17-s − 0.458·19-s − 0.218·21-s − 0.108·23-s + 1/5·25-s + 0.192·27-s − 1.65·29-s − 0.845·31-s − 0.664·33-s − 0.169·35-s + 1.92·37-s + 0.246·39-s − 1.07·41-s − 0.0709·43-s + 0.149·45-s + 0.223·47-s + 1/7·49-s + 0.543·51-s + 0.870·53-s − 0.514·55-s + ⋯

Functional equation

Λ(s)=(1680s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1680s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16801680    =    24⋅3⋅5⋅72^{4} \cdot 3 \cdot 5 \cdot 7
Sign: −1-1
Analytic conductor: 99.123299.1232
Root analytic conductor: 9.956069.95606
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1680, ( :3/2), −1)(2,\ 1680,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
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−pT 1 - p T
7 1+pT 1 + p T
good11 1+42T+p3T2 1 + 42 T + p^{3} T^{2}
13 1−20T+p3T2 1 - 20 T + p^{3} T^{2}
17 1−66T+p3T2 1 - 66 T + p^{3} T^{2}
19 1+2pT+p3T2 1 + 2 p T + p^{3} T^{2}
23 1+12T+p3T2 1 + 12 T + p^{3} T^{2}
29 1+258T+p3T2 1 + 258 T + p^{3} T^{2}
31 1+146T+p3T2 1 + 146 T + p^{3} T^{2}
37 1−434T+p3T2 1 - 434 T + p^{3} T^{2}
41 1+282T+p3T2 1 + 282 T + p^{3} T^{2}
43 1+20T+p3T2 1 + 20 T + p^{3} T^{2}
47 1−72T+p3T2 1 - 72 T + p^{3} T^{2}
53 1−336T+p3T2 1 - 336 T + p^{3} T^{2}
59 1−360T+p3T2 1 - 360 T + p^{3} T^{2}
61 1+682T+p3T2 1 + 682 T + p^{3} T^{2}
67 1+812T+p3T2 1 + 812 T + p^{3} T^{2}
71 1+810T+p3T2 1 + 810 T + p^{3} T^{2}
73 1+124T+p3T2 1 + 124 T + p^{3} T^{2}
79 1+1136T+p3T2 1 + 1136 T + p^{3} T^{2}
83 1+156T+p3T2 1 + 156 T + p^{3} T^{2}
89 1+1038T+p3T2 1 + 1038 T + p^{3} T^{2}
97 1−1208T+p3T2 1 - 1208 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

−8.638307577780672359234010356461, −7.77009943952594211233462479874, −7.21027022986347936511281085244, −6.00533045917114145715677395758, −5.48853096486770319047370649548, −4.31631132439366062534610889767, −3.33312142869497315328442703554, −2.51873453369019607688938975072, −1.47229320978480376234335810280, 0, 1.47229320978480376234335810280, 2.51873453369019607688938975072, 3.33312142869497315328442703554, 4.31631132439366062534610889767, 5.48853096486770319047370649548, 6.00533045917114145715677395758, 7.21027022986347936511281085244, 7.77009943952594211233462479874, 8.638307577780672359234010356461

Graph of the ZZ-function along the critical line