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Properties

Label 2-153-1.1-c3-0-4
Degree 22
Conductor 153153
Sign 11
Analytic cond. 9.027299.02729
Root an. cond. 3.004543.00454
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.56·2-s + 4.72·4-s + 20.6·5-s − 5.24·7-s + 11.6·8-s − 73.6·10-s + 5.21·11-s − 14.8·13-s + 18.7·14-s − 79.4·16-s − 17·17-s + 26.2·19-s + 97.5·20-s − 18.5·22-s + 165.·23-s + 300.·25-s + 52.8·26-s − 24.7·28-s − 42.3·29-s + 263.·31-s + 190.·32-s + 60.6·34-s − 108.·35-s − 322.·37-s − 93.6·38-s + 240.·40-s + 321.·41-s + ⋯
L(s)  = 1  − 1.26·2-s + 0.591·4-s + 1.84·5-s − 0.283·7-s + 0.515·8-s − 2.32·10-s + 0.142·11-s − 0.315·13-s + 0.357·14-s − 1.24·16-s − 0.242·17-s + 0.317·19-s + 1.09·20-s − 0.180·22-s + 1.49·23-s + 2.40·25-s + 0.398·26-s − 0.167·28-s − 0.271·29-s + 1.52·31-s + 1.05·32-s + 0.305·34-s − 0.522·35-s − 1.43·37-s − 0.399·38-s + 0.951·40-s + 1.22·41-s + ⋯

Functional equation

Λ(s)=(153s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(153s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 153153    =    32⋅173^{2} \cdot 17
Sign: 11
Analytic conductor: 9.027299.02729
Root analytic conductor: 3.004543.00454
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 153, ( :3/2), 1)(2,\ 153,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.1567502101.156750210
L(12)L(\frac12) ≈\approx 1.1567502101.156750210
L(52)L(\frac{5}{2}) 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
17 1+17T 1 + 17T
good2 1+3.56T+8T2 1 + 3.56T + 8T^{2}
5 1−20.6T+125T2 1 - 20.6T + 125T^{2}
7 1+5.24T+343T2 1 + 5.24T + 343T^{2}
11 1−5.21T+1.33e3T2 1 - 5.21T + 1.33e3T^{2}
13 1+14.8T+2.19e3T2 1 + 14.8T + 2.19e3T^{2}
19 1−26.2T+6.85e3T2 1 - 26.2T + 6.85e3T^{2}
23 1−165.T+1.21e4T2 1 - 165.T + 1.21e4T^{2}
29 1+42.3T+2.43e4T2 1 + 42.3T + 2.43e4T^{2}
31 1−263.T+2.97e4T2 1 - 263.T + 2.97e4T^{2}
37 1+322.T+5.06e4T2 1 + 322.T + 5.06e4T^{2}
41 1−321.T+6.89e4T2 1 - 321.T + 6.89e4T^{2}
43 1−385.T+7.95e4T2 1 - 385.T + 7.95e4T^{2}
47 1−309.T+1.03e5T2 1 - 309.T + 1.03e5T^{2}
53 1−192.T+1.48e5T2 1 - 192.T + 1.48e5T^{2}
59 1+587.T+2.05e5T2 1 + 587.T + 2.05e5T^{2}
61 1+241.T+2.26e5T2 1 + 241.T + 2.26e5T^{2}
67 1−205.T+3.00e5T2 1 - 205.T + 3.00e5T^{2}
71 1−933.T+3.57e5T2 1 - 933.T + 3.57e5T^{2}
73 1+869.T+3.89e5T2 1 + 869.T + 3.89e5T^{2}
79 1−102.T+4.93e5T2 1 - 102.T + 4.93e5T^{2}
83 1−298.T+5.71e5T2 1 - 298.T + 5.71e5T^{2}
89 1+666.T+7.04e5T2 1 + 666.T + 7.04e5T^{2}
97 1−1.35e3T+9.12e5T2 1 - 1.35e3T + 9.12e5T^{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

−12.61825142944827577183203785288, −10.97289370472511688948647731685, −10.17828974970501294919493201837, −9.391305296833257148221719960173, −8.821769211335353225351863345315, −7.29381656739139693234283352364, −6.24554186892528550864988959631, −4.94720697761208684389158774293, −2.52304543565308784976037146098, −1.14911976625303827165170692907, 1.14911976625303827165170692907, 2.52304543565308784976037146098, 4.94720697761208684389158774293, 6.24554186892528550864988959631, 7.29381656739139693234283352364, 8.821769211335353225351863345315, 9.391305296833257148221719960173, 10.17828974970501294919493201837, 10.97289370472511688948647731685, 12.61825142944827577183203785288

Graph of the ZZ-function along the critical line