↑

Properties

Label 2-3648-1.1-c1-0-51
Degree 22
Conductor 36483648
Sign −1-1
Analytic cond. 29.129429.1294
Root an. cond. 5.397165.39716
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s + 3·7-s + 9-s − 5·11-s + 2·13-s + 15-s − 17-s + 19-s − 3·21-s − 4·23-s − 4·25-s − 27-s + 6·29-s + 10·31-s + 5·33-s − 3·35-s − 2·39-s − 11·43-s − 45-s − 9·47-s + 2·49-s + 51-s − 10·53-s + 5·55-s − 57-s + 4·59-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1.13·7-s + 1/3·9-s − 1.50·11-s + 0.554·13-s + 0.258·15-s − 0.242·17-s + 0.229·19-s − 0.654·21-s − 0.834·23-s − 4/5·25-s − 0.192·27-s + 1.11·29-s + 1.79·31-s + 0.870·33-s − 0.507·35-s − 0.320·39-s − 1.67·43-s − 0.149·45-s − 1.31·47-s + 2/7·49-s + 0.140·51-s − 1.37·53-s + 0.674·55-s − 0.132·57-s + 0.520·59-s + ⋯

Functional equation

Λ(s)=(3648s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(3648s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 36483648    =    26⋅3⋅192^{6} \cdot 3 \cdot 19
Sign: −1-1
Analytic conductor: 29.129429.1294
Root analytic conductor: 5.397165.39716
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 3648, ( :1/2), −1)(2,\ 3648,\ (\ :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)
bad2 1 1
3 1+T 1 + T
19 1−T 1 - T
good5 1+T+pT2 1 + T + p T^{2}
7 1−3T+pT2 1 - 3 T + p T^{2}
11 1+5T+pT2 1 + 5 T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1+T+pT2 1 + T + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1−10T+pT2 1 - 10 T + p T^{2}
37 1+pT2 1 + p T^{2}
41 1+pT2 1 + p T^{2}
43 1+11T+pT2 1 + 11 T + p T^{2}
47 1+9T+pT2 1 + 9 T + p T^{2}
53 1+10T+pT2 1 + 10 T + p T^{2}
59 1−4T+pT2 1 - 4 T + p T^{2}
61 1−5T+pT2 1 - 5 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1+8T+pT2 1 + 8 T + p T^{2}
73 1−13T+pT2 1 - 13 T + p T^{2}
79 1+4T+pT2 1 + 4 T + p T^{2}
83 1+4T+pT2 1 + 4 T + p T^{2}
89 1+6T+pT2 1 + 6 T + p T^{2}
97 1−2T+pT2 1 - 2 T + p 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

−8.137259653188121390855447482027, −7.65627920571478032514336271694, −6.60850791131037206491890289680, −5.90641506250154868855227026945, −4.90222652055451670072407703452, −4.68719487779910370709247985620, −3.52150549884707699550976993427, −2.43726074335480144289137857493, −1.36309445163470127656106423645, 0, 1.36309445163470127656106423645, 2.43726074335480144289137857493, 3.52150549884707699550976993427, 4.68719487779910370709247985620, 4.90222652055451670072407703452, 5.90641506250154868855227026945, 6.60850791131037206491890289680, 7.65627920571478032514336271694, 8.137259653188121390855447482027

Graph of the ZZ-function along the critical line