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Properties

Label 2-55-55.54-c16-0-74
Degree 22
Conductor 5555
Sign 11
Analytic cond. 89.278489.2784
Root an. cond. 9.448739.44873
Motivic weight 1616
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 17·2-s − 6.52e4·4-s + 3.90e5·5-s + 9.88e6·7-s + 2.22e6·8-s + 4.30e7·9-s − 6.64e6·10-s + 2.14e8·11-s + 6.78e8·13-s − 1.68e8·14-s + 4.23e9·16-s + 1.39e10·17-s − 7.31e8·18-s − 2.54e10·20-s − 3.64e9·22-s + 1.52e11·25-s − 1.15e10·26-s − 6.45e11·28-s − 1.20e12·31-s − 2.17e11·32-s − 2.36e11·34-s + 3.86e12·35-s − 2.80e12·36-s + 8.68e11·40-s − 7.61e12·43-s − 1.39e13·44-s + 1.68e13·45-s + ⋯
L(s)  = 1  − 0.0664·2-s − 0.995·4-s + 5-s + 1.71·7-s + 0.132·8-s + 9-s − 0.0664·10-s + 11-s + 0.831·13-s − 0.113·14-s + 0.986·16-s + 1.99·17-s − 0.0664·18-s − 0.995·20-s − 0.0664·22-s + 25-s − 0.0551·26-s − 1.70·28-s − 1.41·31-s − 0.198·32-s − 0.132·34-s + 1.71·35-s − 0.995·36-s + 0.132·40-s − 0.651·43-s − 0.995·44-s + 45-s + ⋯

Functional equation

Λ(s)=(55s/2 ΓC(s) L(s)=( Λ(17−s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(17-s) \end{aligned}
Λ(s)=(55s/2 ΓC(s+8) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5555    =    5⋅115 \cdot 11
Sign: 11
Analytic conductor: 89.278489.2784
Root analytic conductor: 9.448739.44873
Motivic weight: 1616
Rational: yes
Arithmetic: yes
Character: χ55(54,⋅)\chi_{55} (54, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 55, ( :8), 1)(2,\ 55,\ (\ :8),\ 1)

Particular Values

L(172)L(\frac{17}{2}) ≈\approx 3.6042699093.604269909
L(12)L(\frac12) ≈\approx 3.6042699093.604269909
L(9)L(9) 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)
bad5 1−p8T 1 - p^{8} T
11 1−p8T 1 - p^{8} T
good2 1+17T+p16T2 1 + 17 T + p^{16} T^{2}
3 (1−p8T)(1+p8T) ( 1 - p^{8} T )( 1 + p^{8} T )
7 1−9886078T+p16T2 1 - 9886078 T + p^{16} T^{2}
13 1−678010558T+p16T2 1 - 678010558 T + p^{16} T^{2}
17 1−13921943038T+p16T2 1 - 13921943038 T + p^{16} T^{2}
19 (1−p8T)(1+p8T) ( 1 - p^{8} T )( 1 + p^{8} T )
23 (1−p8T)(1+p8T) ( 1 - p^{8} T )( 1 + p^{8} T )
29 (1−p8T)(1+p8T) ( 1 - p^{8} T )( 1 + p^{8} T )
31 1+1206552215038T+p16T2 1 + 1206552215038 T + p^{16} T^{2}
37 (1−p8T)(1+p8T) ( 1 - p^{8} T )( 1 + p^{8} T )
41 (1−p8T)(1+p8T) ( 1 - p^{8} T )( 1 + p^{8} T )
43 1+7613774646722T+p16T2 1 + 7613774646722 T + p^{16} T^{2}
47 (1−p8T)(1+p8T) ( 1 - p^{8} T )( 1 + p^{8} T )
53 (1−p8T)(1+p8T) ( 1 - p^{8} T )( 1 + p^{8} T )
59 1+140214236988478T+p16T2 1 + 140214236988478 T + p^{16} T^{2}
61 (1−p8T)(1+p8T) ( 1 - p^{8} T )( 1 + p^{8} T )
67 (1−p8T)(1+p8T) ( 1 - p^{8} T )( 1 + p^{8} T )
71 1+77726196639358T+p16T2 1 + 77726196639358 T + p^{16} T^{2}
73 1+1564720076407682T+p16T2 1 + 1564720076407682 T + p^{16} T^{2}
79 (1−p8T)(1+p8T) ( 1 - p^{8} T )( 1 + p^{8} T )
83 1+3613022253130562T+p16T2 1 + 3613022253130562 T + p^{16} T^{2}
89 1−7841390882244482T+p16T2 1 - 7841390882244482 T + p^{16} T^{2}
97 (1−p8T)(1+p8T) ( 1 - p^{8} T )( 1 + p^{8} 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

−12.07937498799510422097805548968, −10.65882628918100187257894316811, −9.647501157488506052792719638095, −8.642864920731345997177030570406, −7.51254922227420762832984551575, −5.78057515393475360659004218140, −4.83717085441055109790416199158, −3.71290232737277187300001029010, −1.46745121703469827642332011530, −1.25877530102447357949494549842, 1.25877530102447357949494549842, 1.46745121703469827642332011530, 3.71290232737277187300001029010, 4.83717085441055109790416199158, 5.78057515393475360659004218140, 7.51254922227420762832984551575, 8.642864920731345997177030570406, 9.647501157488506052792719638095, 10.65882628918100187257894316811, 12.07937498799510422097805548968

Graph of the ZZ-function along the critical line