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Properties

Label 2-1216-19.18-c2-0-59
Degree 22
Conductor 12161216
Sign 11
Analytic cond. 33.133633.1336
Root an. cond. 5.756175.75617
Motivic weight 22
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9·5-s + 5·7-s + 9·9-s + 3·11-s + 15·17-s − 19·19-s + 30·23-s + 56·25-s + 45·35-s − 85·43-s + 81·45-s − 75·47-s − 24·49-s + 27·55-s − 103·61-s + 45·63-s − 25·73-s + 15·77-s + 81·81-s + 90·83-s + 135·85-s − 171·95-s + 27·99-s + 102·101-s + 270·115-s + 75·119-s + ⋯
L(s)  = 1  + 9/5·5-s + 5/7·7-s + 9-s + 3/11·11-s + 0.882·17-s − 19-s + 1.30·23-s + 2.23·25-s + 9/7·35-s − 1.97·43-s + 9/5·45-s − 1.59·47-s − 0.489·49-s + 0.490·55-s − 1.68·61-s + 5/7·63-s − 0.342·73-s + 0.194·77-s + 81-s + 1.08·83-s + 1.58·85-s − 9/5·95-s + 3/11·99-s + 1.00·101-s + 2.34·115-s + 0.630·119-s + ⋯

Functional equation

Λ(s)=(1216s/2 ΓC(s) L(s)=( Λ(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}
Λ(s)=(1216s/2 ΓC(s+1) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12161216    =    26⋅192^{6} \cdot 19
Sign: 11
Analytic conductor: 33.133633.1336
Root analytic conductor: 5.756175.75617
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: χ1216(1025,⋅)\chi_{1216} (1025, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1216, ( :1), 1)(2,\ 1216,\ (\ :1),\ 1)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 3.4808661593.480866159
L(12)L(\frac12) ≈\approx 3.4808661593.480866159
L(2)L(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
19 1+pT 1 + p T
good3 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
5 1−9T+p2T2 1 - 9 T + p^{2} T^{2}
7 1−5T+p2T2 1 - 5 T + p^{2} T^{2}
11 1−3T+p2T2 1 - 3 T + p^{2} T^{2}
13 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
17 1−15T+p2T2 1 - 15 T + p^{2} T^{2}
23 1−30T+p2T2 1 - 30 T + p^{2} T^{2}
29 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
31 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
37 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
41 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
43 1+85T+p2T2 1 + 85 T + p^{2} T^{2}
47 1+75T+p2T2 1 + 75 T + p^{2} T^{2}
53 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
59 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
61 1+103T+p2T2 1 + 103 T + p^{2} T^{2}
67 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
71 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
73 1+25T+p2T2 1 + 25 T + p^{2} T^{2}
79 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
83 1−90T+p2T2 1 - 90 T + p^{2} T^{2}
89 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
97 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
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   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

−9.617769548368163634254218158588, −8.905780799669726346291639114102, −7.950721780379256626396910939638, −6.84561280985131032950905934575, −6.28064280890198498089936432599, −5.21575972898535996476804658227, −4.66152478919190597730094087242, −3.19587303235678236099511865911, −1.90008376656696799905281197083, −1.32086676398853590177841162574, 1.32086676398853590177841162574, 1.90008376656696799905281197083, 3.19587303235678236099511865911, 4.66152478919190597730094087242, 5.21575972898535996476804658227, 6.28064280890198498089936432599, 6.84561280985131032950905934575, 7.950721780379256626396910939638, 8.905780799669726346291639114102, 9.617769548368163634254218158588

Graph of the ZZ-function along the critical line