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Properties

Label 2-59-59.58-c18-0-38
Degree 22
Conductor 5959
Sign 11
Analytic cond. 121.177121.177
Root an. cond. 11.008011.0080
Motivic weight 1818
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.08e4·3-s + 2.62e5·4-s − 1.98e6·5-s − 5.09e7·7-s − 2.70e8·9-s + 2.83e9·12-s − 2.14e10·15-s + 6.87e10·16-s − 2.16e11·17-s − 6.24e10·19-s − 5.20e11·20-s − 5.51e11·21-s + 1.26e11·25-s − 7.11e12·27-s − 1.33e13·28-s + 2.89e13·29-s + 1.01e14·35-s − 7.09e13·36-s + 6.52e14·41-s + 5.37e14·45-s + 7.42e14·48-s + 9.70e14·49-s − 2.34e15·51-s + 5.73e15·53-s − 6.74e14·57-s − 8.66e15·59-s − 5.62e15·60-s + ⋯
L(s)  = 1  + 0.549·3-s + 4-s − 1.01·5-s − 1.26·7-s − 0.698·9-s + 0.549·12-s − 0.558·15-s + 16-s − 1.82·17-s − 0.193·19-s − 1.01·20-s − 0.693·21-s + 0.0331·25-s − 0.932·27-s − 1.26·28-s + 1.99·29-s + 1.28·35-s − 0.698·36-s + 1.99·41-s + 0.709·45-s + 0.549·48-s + 0.596·49-s − 1.00·51-s + 1.73·53-s − 0.106·57-s − 59-s − 0.558·60-s + ⋯

Functional equation

Λ(s)=(59s/2 ΓC(s) L(s)=( Λ(19−s)\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(19-s) \end{aligned}
Λ(s)=(59s/2 ΓC(s+9) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5959
Sign: 11
Analytic conductor: 121.177121.177
Root analytic conductor: 11.008011.0080
Motivic weight: 1818
Rational: yes
Arithmetic: yes
Character: χ59(58,⋅)\chi_{59} (58, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 59, ( :9), 1)(2,\ 59,\ (\ :9),\ 1)

Particular Values

L(192)L(\frac{19}{2}) ≈\approx 1.5696984361.569698436
L(12)L(\frac12) ≈\approx 1.5696984361.569698436
L(10)L(10) 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)
bad59 1+p9T 1 + p^{9} T
good2 (1−p9T)(1+p9T) ( 1 - p^{9} T )( 1 + p^{9} T )
3 1−10810T+p18T2 1 - 10810 T + p^{18} T^{2}
5 1+1985254T+p18T2 1 + 1985254 T + p^{18} T^{2}
7 1+50982910T+p18T2 1 + 50982910 T + p^{18} T^{2}
11 (1−p9T)(1+p9T) ( 1 - p^{9} T )( 1 + p^{9} T )
13 (1−p9T)(1+p9T) ( 1 - p^{9} T )( 1 + p^{9} T )
17 1+216651752350T+p18T2 1 + 216651752350 T + p^{18} T^{2}
19 1+62437037542T+p18T2 1 + 62437037542 T + p^{18} T^{2}
23 (1−p9T)(1+p9T) ( 1 - p^{9} T )( 1 + p^{9} T )
29 1−28956785336138T+p18T2 1 - 28956785336138 T + p^{18} T^{2}
31 (1−p9T)(1+p9T) ( 1 - p^{9} T )( 1 + p^{9} T )
37 (1−p9T)(1+p9T) ( 1 - p^{9} T )( 1 + p^{9} T )
41 1−652243002714578T+p18T2 1 - 652243002714578 T + p^{18} T^{2}
43 (1−p9T)(1+p9T) ( 1 - p^{9} T )( 1 + p^{9} T )
47 (1−p9T)(1+p9T) ( 1 - p^{9} T )( 1 + p^{9} T )
53 1−5739806619558650T+p18T2 1 - 5739806619558650 T + p^{18} T^{2}
61 (1−p9T)(1+p9T) ( 1 - p^{9} T )( 1 + p^{9} T )
67 (1−p9T)(1+p9T) ( 1 - p^{9} T )( 1 + p^{9} T )
71 1+56563270329694462T+p18T2 1 + 56563270329694462 T + p^{18} T^{2}
73 (1−p9T)(1+p9T) ( 1 - p^{9} T )( 1 + p^{9} T )
79 1−186548867397995762T+p18T2 1 - 186548867397995762 T + p^{18} T^{2}
83 (1−p9T)(1+p9T) ( 1 - p^{9} T )( 1 + p^{9} T )
89 (1−p9T)(1+p9T) ( 1 - p^{9} T )( 1 + p^{9} T )
97 (1−p9T)(1+p9T) ( 1 - p^{9} T )( 1 + p^{9} 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

−11.53295753629708916888933910962, −10.56612086627466476731861376004, −9.129414633071559297767085000262, −8.088725262019185802361117462785, −6.93897317884380294862650413420, −6.09591468881936247563605981003, −4.17580313281605674370252671609, −3.06037902721396444993818262132, −2.36257247690574953730209512152, −0.52675109206253678704580272709, 0.52675109206253678704580272709, 2.36257247690574953730209512152, 3.06037902721396444993818262132, 4.17580313281605674370252671609, 6.09591468881936247563605981003, 6.93897317884380294862650413420, 8.088725262019185802361117462785, 9.129414633071559297767085000262, 10.56612086627466476731861376004, 11.53295753629708916888933910962

Graph of the ZZ-function along the critical line