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Properties

Label 2-55-55.54-c8-0-21
Degree 22
Conductor 5555
Sign 11
Analytic cond. 22.405822.4058
Root an. cond. 4.733474.73347
Motivic weight 88
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 23·2-s + 273·4-s + 625·5-s + 1.28e3·7-s − 391·8-s + 6.56e3·9-s − 1.43e4·10-s + 1.46e4·11-s − 3.08e4·13-s − 2.94e4·14-s − 6.08e4·16-s − 5.43e3·17-s − 1.50e5·18-s + 1.70e5·20-s − 3.36e5·22-s + 3.90e5·25-s + 7.10e5·26-s + 3.49e5·28-s + 7.06e5·31-s + 1.50e6·32-s + 1.25e5·34-s + 8.01e5·35-s + 1.79e6·36-s − 2.44e5·40-s + 5.56e6·43-s + 3.99e6·44-s + 4.10e6·45-s + ⋯
L(s)  = 1  − 1.43·2-s + 1.06·4-s + 5-s + 0.533·7-s − 0.0954·8-s + 9-s − 1.43·10-s + 11-s − 1.08·13-s − 0.767·14-s − 0.929·16-s − 0.0651·17-s − 1.43·18-s + 1.06·20-s − 1.43·22-s + 25-s + 1.55·26-s + 0.569·28-s + 0.765·31-s + 1.43·32-s + 0.0935·34-s + 0.533·35-s + 1.06·36-s − 0.0954·40-s + 1.62·43-s + 1.06·44-s + 45-s + ⋯

Functional equation

Λ(s)=(55s/2 ΓC(s) L(s)=( Λ(9−s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}
Λ(s)=(55s/2 ΓC(s+4) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5555    =    5⋅115 \cdot 11
Sign: 11
Analytic conductor: 22.405822.4058
Root analytic conductor: 4.733474.73347
Motivic weight: 88
Rational: yes
Arithmetic: yes
Character: χ55(54,⋅)\chi_{55} (54, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 55, ( :4), 1)(2,\ 55,\ (\ :4),\ 1)

Particular Values

L(92)L(\frac{9}{2}) ≈\approx 1.3060232191.306023219
L(12)L(\frac12) ≈\approx 1.3060232191.306023219
L(5)L(5) 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−p4T 1 - p^{4} T
11 1−p4T 1 - p^{4} T
good2 1+23T+p8T2 1 + 23 T + p^{8} T^{2}
3 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} T )
7 1−1282T+p8T2 1 - 1282 T + p^{8} T^{2}
13 1+30878T+p8T2 1 + 30878 T + p^{8} T^{2}
17 1+5438T+p8T2 1 + 5438 T + p^{8} T^{2}
19 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} T )
23 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} T )
29 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} T )
31 1−706562T+p8T2 1 - 706562 T + p^{8} T^{2}
37 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} T )
41 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} T )
43 1−5566882T+p8T2 1 - 5566882 T + p^{8} T^{2}
47 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} T )
53 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} T )
59 1+12387358T+p8T2 1 + 12387358 T + p^{8} T^{2}
61 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} T )
67 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} T )
71 1+34839358T+p8T2 1 + 34839358 T + p^{8} T^{2}
73 1−56370562T+p8T2 1 - 56370562 T + p^{8} T^{2}
79 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} T )
83 1−90097762T+p8T2 1 - 90097762 T + p^{8} T^{2}
89 1−125357762T+p8T2 1 - 125357762 T + p^{8} T^{2}
97 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} 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

−13.66753412069754556491935671653, −12.24400623577549414426302551860, −10.79307235445606406743539395895, −9.801606776071171627755142298209, −9.127689559007839723887304524961, −7.68438297925489974175425333354, −6.57352326877886123066920995327, −4.66480089888989234981502317529, −2.08299614635532685606586800263, −1.03027297756288283171345149524, 1.03027297756288283171345149524, 2.08299614635532685606586800263, 4.66480089888989234981502317529, 6.57352326877886123066920995327, 7.68438297925489974175425333354, 9.127689559007839723887304524961, 9.801606776071171627755142298209, 10.79307235445606406743539395895, 12.24400623577549414426302551860, 13.66753412069754556491935671653

Graph of the ZZ-function along the critical line