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Properties

Label 2-11-11.10-c32-0-16
Degree 22
Conductor 1111
Sign 11
Analytic cond. 71.353371.3533
Root an. cond. 8.447088.44708
Motivic weight 3232
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8.59e7·3-s + 4.29e9·4-s − 9.72e9·5-s + 5.53e15·9-s + 4.59e16·11-s − 3.69e17·12-s + 8.36e17·15-s + 1.84e19·16-s − 4.17e19·20-s + 3.65e21·23-s − 2.31e22·25-s − 3.16e23·27-s − 1.00e24·31-s − 3.95e24·33-s + 2.37e25·36-s + 1.77e25·37-s + 1.97e26·44-s − 5.38e25·45-s − 1.02e27·47-s − 1.58e27·48-s + 1.10e27·49-s + 4.02e27·53-s − 4.47e26·55-s + 3.35e28·59-s + 3.59e27·60-s + 7.92e28·64-s − 1.53e29·67-s + ⋯
L(s)  = 1  − 1.99·3-s + 4-s − 0.0637·5-s + 2.98·9-s + 11-s − 1.99·12-s + 0.127·15-s + 16-s − 0.0637·20-s + 0.596·23-s − 0.995·25-s − 3.97·27-s − 1.38·31-s − 1.99·33-s + 2.98·36-s + 1.43·37-s + 44-s − 0.190·45-s − 1.80·47-s − 1.99·48-s + 49-s + 1.03·53-s − 0.0637·55-s + 1.55·59-s + 0.127·60-s + 64-s − 0.931·67-s + ⋯

Functional equation

Λ(s)=(11s/2 ΓC(s) L(s)=( Λ(33−s)\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(33-s) \end{aligned}
Λ(s)=(11s/2 ΓC(s+16) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1111
Sign: 11
Analytic conductor: 71.353371.3533
Root analytic conductor: 8.447088.44708
Motivic weight: 3232
Rational: yes
Arithmetic: yes
Character: χ11(10,⋅)\chi_{11} (10, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 11, ( :16), 1)(2,\ 11,\ (\ :16),\ 1)

Particular Values

L(332)L(\frac{33}{2}) ≈\approx 1.4649939451.464993945
L(12)L(\frac12) ≈\approx 1.4649939451.464993945
L(17)L(17) 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)
bad11 1−p16T 1 - p^{16} T
good2 (1−p16T)(1+p16T) ( 1 - p^{16} T )( 1 + p^{16} T )
3 1+85968833T+p32T2 1 + 85968833 T + p^{32} T^{2}
5 1+9728091649T+p32T2 1 + 9728091649 T + p^{32} T^{2}
7 (1−p16T)(1+p16T) ( 1 - p^{16} T )( 1 + p^{16} T )
13 (1−p16T)(1+p16T) ( 1 - p^{16} T )( 1 + p^{16} T )
17 (1−p16T)(1+p16T) ( 1 - p^{16} T )( 1 + p^{16} T )
19 (1−p16T)(1+p16T) ( 1 - p^{16} T )( 1 + p^{16} T )
23 1− 1 - 36 ⁣⋯ ⁣4736\!\cdots\!47T+p32T2 T + p^{32} T^{2}
29 (1−p16T)(1+p16T) ( 1 - p^{16} T )( 1 + p^{16} T )
31 1+ 1 + 10 ⁣⋯ ⁣1310\!\cdots\!13T+p32T2 T + p^{32} T^{2}
37 1− 1 - 17 ⁣⋯ ⁣0717\!\cdots\!07T+p32T2 T + p^{32} T^{2}
41 (1−p16T)(1+p16T) ( 1 - p^{16} T )( 1 + p^{16} T )
43 (1−p16T)(1+p16T) ( 1 - p^{16} T )( 1 + p^{16} T )
47 1+ 1 + 10 ⁣⋯ ⁣5810\!\cdots\!58T+p32T2 T + p^{32} T^{2}
53 1− 1 - 40 ⁣⋯ ⁣4240\!\cdots\!42T+p32T2 T + p^{32} T^{2}
59 1− 1 - 33 ⁣⋯ ⁣0733\!\cdots\!07T+p32T2 T + p^{32} T^{2}
61 (1−p16T)(1+p16T) ( 1 - p^{16} T )( 1 + p^{16} T )
67 1+ 1 + 15 ⁣⋯ ⁣1315\!\cdots\!13T+p32T2 T + p^{32} T^{2}
71 1− 1 - 70 ⁣⋯ ⁣6770\!\cdots\!67T+p32T2 T + p^{32} T^{2}
73 (1−p16T)(1+p16T) ( 1 - p^{16} T )( 1 + p^{16} T )
79 (1−p16T)(1+p16T) ( 1 - p^{16} T )( 1 + p^{16} T )
83 (1−p16T)(1+p16T) ( 1 - p^{16} T )( 1 + p^{16} T )
89 1+ 1 + 30 ⁣⋯ ⁣5330\!\cdots\!53T+p32T2 T + p^{32} T^{2}
97 1+ 1 + 40 ⁣⋯ ⁣3340\!\cdots\!33T+p32T2 T + p^{32} T^{2}
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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

−12.79681991640641744096439676747, −11.69815852947917030429194207633, −11.12793170999355942816611734265, −9.862825479993612420324049217911, −7.31160544412265240312694550235, −6.40906376990926192762832537484, −5.46735563983969898430666425266, −3.96812195673020274630076984686, −1.75624228896501795476104167445, −0.72381146104502178281250642626, 0.72381146104502178281250642626, 1.75624228896501795476104167445, 3.96812195673020274630076984686, 5.46735563983969898430666425266, 6.40906376990926192762832537484, 7.31160544412265240312694550235, 9.862825479993612420324049217911, 11.12793170999355942816611734265, 11.69815852947917030429194207633, 12.79681991640641744096439676747

Graph of the ZZ-function along the critical line