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Properties

Label 2-207-23.22-c6-0-31
Degree 22
Conductor 207207
Sign 11
Analytic cond. 47.621147.6211
Root an. cond. 6.900816.90081
Motivic weight 66
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7·2-s − 15·4-s − 553·8-s + 1.08e3·13-s − 2.91e3·16-s + 1.21e4·23-s + 1.56e4·25-s + 7.57e3·26-s − 3.07e4·29-s + 5.87e4·31-s + 1.50e4·32-s − 4.36e4·41-s + 8.51e4·46-s + 2.05e5·47-s + 1.17e5·49-s + 1.09e5·50-s − 1.62e4·52-s − 2.15e5·58-s + 2.53e5·59-s + 4.11e5·62-s + 2.91e5·64-s − 6.67e5·71-s + 7.25e5·73-s − 3.05e5·82-s − 1.82e5·92-s + 1.43e6·94-s + 8.23e5·98-s + ⋯
L(s)  = 1  + 7/8·2-s − 0.234·4-s − 1.08·8-s + 0.492·13-s − 0.710·16-s + 23-s + 25-s + 0.430·26-s − 1.26·29-s + 1.97·31-s + 0.458·32-s − 0.633·41-s + 7/8·46-s + 1.97·47-s + 49-s + 7/8·50-s − 0.115·52-s − 1.10·58-s + 1.23·59-s + 1.72·62-s + 1.11·64-s − 1.86·71-s + 1.86·73-s − 0.553·82-s − 0.234·92-s + 1.73·94-s + 7/8·98-s + ⋯

Functional equation

Λ(s)=(207s/2 ΓC(s) L(s)=( Λ(7−s)\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}
Λ(s)=(207s/2 ΓC(s+3) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 207207    =    32⋅233^{2} \cdot 23
Sign: 11
Analytic conductor: 47.621147.6211
Root analytic conductor: 6.900816.90081
Motivic weight: 66
Rational: yes
Arithmetic: yes
Character: χ207(91,⋅)\chi_{207} (91, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 207, ( :3), 1)(2,\ 207,\ (\ :3),\ 1)

Particular Values

L(72)L(\frac{7}{2}) ≈\approx 2.6913057302.691305730
L(12)L(\frac12) ≈\approx 2.6913057302.691305730
L(4)L(4) 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)
bad3 1 1
23 1−p3T 1 - p^{3} T
good2 1−7T+p6T2 1 - 7 T + p^{6} T^{2}
5 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
7 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
11 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
13 1−1082T+p6T2 1 - 1082 T + p^{6} T^{2}
17 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
19 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
29 1+30746T+p6T2 1 + 30746 T + p^{6} T^{2}
31 1−58754T+p6T2 1 - 58754 T + p^{6} T^{2}
37 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
41 1+43634T+p6T2 1 + 43634 T + p^{6} T^{2}
43 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
47 1−205342T+p6T2 1 - 205342 T + p^{6} T^{2}
53 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
59 1−253942T+p6T2 1 - 253942 T + p^{6} T^{2}
61 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
67 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
71 1+667154T+p6T2 1 + 667154 T + p^{6} T^{2}
73 1−725042T+p6T2 1 - 725042 T + p^{6} T^{2}
79 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
83 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
89 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
97 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} 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

−11.54716265197296038714606853706, −10.45554556745774000630636596661, −9.237354197399198739775347048622, −8.452792376307919635516206183209, −7.01084387251819033234980784863, −5.89538152077761041911763273143, −4.90776775002447377488476556734, −3.85443700897885935858428715053, −2.71077784958008177488851218020, −0.831458597353747937277135148967, 0.831458597353747937277135148967, 2.71077784958008177488851218020, 3.85443700897885935858428715053, 4.90776775002447377488476556734, 5.89538152077761041911763273143, 7.01084387251819033234980784863, 8.452792376307919635516206183209, 9.237354197399198739775347048622, 10.45554556745774000630636596661, 11.54716265197296038714606853706

Graph of the ZZ-function along the critical line