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Properties

Label 2-153-1.1-c3-0-13
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  + 5.03·2-s + 17.3·4-s − 0.885·5-s + 3.81·7-s + 46.9·8-s − 4.45·10-s + 52.3·11-s − 8.06·13-s + 19.2·14-s + 97.5·16-s + 17·17-s − 66.5·19-s − 15.3·20-s + 263.·22-s − 180.·23-s − 124.·25-s − 40.5·26-s + 66.1·28-s + 41.2·29-s − 34.9·31-s + 115.·32-s + 85.5·34-s − 3.38·35-s + 130.·37-s − 334.·38-s − 41.5·40-s + 17.9·41-s + ⋯
L(s)  = 1  + 1.77·2-s + 2.16·4-s − 0.0792·5-s + 0.206·7-s + 2.07·8-s − 0.140·10-s + 1.43·11-s − 0.171·13-s + 0.366·14-s + 1.52·16-s + 0.242·17-s − 0.803·19-s − 0.171·20-s + 2.55·22-s − 1.63·23-s − 0.993·25-s − 0.305·26-s + 0.446·28-s + 0.264·29-s − 0.202·31-s + 0.638·32-s + 0.431·34-s − 0.0163·35-s + 0.579·37-s − 1.42·38-s − 0.164·40-s + 0.0682·41-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 4.6606202204.660620220
L(12)L(\frac12) ≈\approx 4.6606202204.660620220
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 1−5.03T+8T2 1 - 5.03T + 8T^{2}
5 1+0.885T+125T2 1 + 0.885T + 125T^{2}
7 1−3.81T+343T2 1 - 3.81T + 343T^{2}
11 1−52.3T+1.33e3T2 1 - 52.3T + 1.33e3T^{2}
13 1+8.06T+2.19e3T2 1 + 8.06T + 2.19e3T^{2}
19 1+66.5T+6.85e3T2 1 + 66.5T + 6.85e3T^{2}
23 1+180.T+1.21e4T2 1 + 180.T + 1.21e4T^{2}
29 1−41.2T+2.43e4T2 1 - 41.2T + 2.43e4T^{2}
31 1+34.9T+2.97e4T2 1 + 34.9T + 2.97e4T^{2}
37 1−130.T+5.06e4T2 1 - 130.T + 5.06e4T^{2}
41 1−17.9T+6.89e4T2 1 - 17.9T + 6.89e4T^{2}
43 1−277.T+7.95e4T2 1 - 277.T + 7.95e4T^{2}
47 1+463.T+1.03e5T2 1 + 463.T + 1.03e5T^{2}
53 1−329.T+1.48e5T2 1 - 329.T + 1.48e5T^{2}
59 1+678.T+2.05e5T2 1 + 678.T + 2.05e5T^{2}
61 1−340.T+2.26e5T2 1 - 340.T + 2.26e5T^{2}
67 1−15.3T+3.00e5T2 1 - 15.3T + 3.00e5T^{2}
71 1−670.T+3.57e5T2 1 - 670.T + 3.57e5T^{2}
73 1−193.T+3.89e5T2 1 - 193.T + 3.89e5T^{2}
79 1−1.08e3T+4.93e5T2 1 - 1.08e3T + 4.93e5T^{2}
83 1−865.T+5.71e5T2 1 - 865.T + 5.71e5T^{2}
89 1+1.12e3T+7.04e5T2 1 + 1.12e3T + 7.04e5T^{2}
97 1+379.T+9.12e5T2 1 + 379.T + 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

−12.48489624373236775736123338260, −11.88052770256101306632685248067, −11.02737283016101839651636562940, −9.624396572606530069461395954809, −8.002435324201369610023989484499, −6.65283483138256497126441004186, −5.87803139257014233075472377117, −4.47662001400708767625410092592, −3.66187611369757625938922168409, −1.98678942895908199373291215876, 1.98678942895908199373291215876, 3.66187611369757625938922168409, 4.47662001400708767625410092592, 5.87803139257014233075472377117, 6.65283483138256497126441004186, 8.002435324201369610023989484499, 9.624396572606530069461395954809, 11.02737283016101839651636562940, 11.88052770256101306632685248067, 12.48489624373236775736123338260

Graph of the ZZ-function along the critical line