↑

Properties

Label 2-1638-1.1-c1-0-13
Degree 22
Conductor 16381638
Sign 11
Analytic cond. 13.079413.0794
Root an. cond. 3.616553.61655
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 7-s + 8-s + 5·11-s − 13-s + 14-s + 16-s + 4·17-s + 2·19-s + 5·22-s − 5·23-s − 5·25-s − 26-s + 28-s − 4·29-s + 31-s + 32-s + 4·34-s + 7·37-s + 2·38-s + 9·41-s − 12·43-s + 5·44-s − 5·46-s + 7·47-s + 49-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s + 1.50·11-s − 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.970·17-s + 0.458·19-s + 1.06·22-s − 1.04·23-s − 25-s − 0.196·26-s + 0.188·28-s − 0.742·29-s + 0.179·31-s + 0.176·32-s + 0.685·34-s + 1.15·37-s + 0.324·38-s + 1.40·41-s − 1.82·43-s + 0.753·44-s − 0.737·46-s + 1.02·47-s + 1/7·49-s + ⋯

Functional equation

Λ(s)=(1638s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1638s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16381638    =    2⋅32⋅7⋅132 \cdot 3^{2} \cdot 7 \cdot 13
Sign: 11
Analytic conductor: 13.079413.0794
Root analytic conductor: 3.616553.61655
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1638, ( :1/2), 1)(2,\ 1638,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.0934200623.093420062
L(12)L(\frac12) ≈\approx 3.0934200623.093420062
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−T 1 - T
3 1 1
7 1−T 1 - T
13 1+T 1 + T
good5 1+pT2 1 + p T^{2}
11 1−5T+pT2 1 - 5 T + p T^{2}
17 1−4T+pT2 1 - 4 T + p T^{2}
19 1−2T+pT2 1 - 2 T + p T^{2}
23 1+5T+pT2 1 + 5 T + p T^{2}
29 1+4T+pT2 1 + 4 T + p T^{2}
31 1−T+pT2 1 - T + p T^{2}
37 1−7T+pT2 1 - 7 T + p T^{2}
41 1−9T+pT2 1 - 9 T + p T^{2}
43 1+12T+pT2 1 + 12 T + p T^{2}
47 1−7T+pT2 1 - 7 T + p T^{2}
53 1−4T+pT2 1 - 4 T + p T^{2}
59 1−6T+pT2 1 - 6 T + p T^{2}
61 1−13T+pT2 1 - 13 T + p T^{2}
67 1−11T+pT2 1 - 11 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1−7T+pT2 1 - 7 T + p T^{2}
79 1+17T+pT2 1 + 17 T + p T^{2}
83 1+4T+pT2 1 + 4 T + p T^{2}
89 1+14T+pT2 1 + 14 T + p T^{2}
97 1−5T+pT2 1 - 5 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

−9.612571086844589429454518424632, −8.469152938024698877370110317561, −7.68409626813723475327909628947, −6.90643848137944117202320139495, −5.98244664513634320441588551536, −5.36737041037488579734906212522, −4.15982302022081941279948742151, −3.71179494687009087455073957292, −2.36943591207239051047957475876, −1.23766611312278004155286222469, 1.23766611312278004155286222469, 2.36943591207239051047957475876, 3.71179494687009087455073957292, 4.15982302022081941279948742151, 5.36737041037488579734906212522, 5.98244664513634320441588551536, 6.90643848137944117202320139495, 7.68409626813723475327909628947, 8.469152938024698877370110317561, 9.612571086844589429454518424632

Graph of the ZZ-function along the critical line