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Properties

Label 2-153-1.1-c3-0-17
Degree 22
Conductor 153153
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.55·2-s − 1.47·4-s − 8.47·5-s + 3.66·7-s − 24.1·8-s − 21.6·10-s − 61.4·11-s + 20.9·13-s + 9.35·14-s − 50.0·16-s − 17·17-s − 102.·19-s + 12.4·20-s − 157.·22-s + 27.1·23-s − 53.2·25-s + 53.6·26-s − 5.38·28-s + 145.·29-s + 72.0·31-s + 65.6·32-s − 43.4·34-s − 31.0·35-s + 371.·37-s − 262.·38-s + 204.·40-s − 348.·41-s + ⋯
L(s)  = 1  + 0.903·2-s − 0.183·4-s − 0.757·5-s + 0.197·7-s − 1.06·8-s − 0.684·10-s − 1.68·11-s + 0.447·13-s + 0.178·14-s − 0.782·16-s − 0.242·17-s − 1.24·19-s + 0.139·20-s − 1.52·22-s + 0.246·23-s − 0.425·25-s + 0.404·26-s − 0.0363·28-s + 0.930·29-s + 0.417·31-s + 0.362·32-s − 0.219·34-s − 0.149·35-s + 1.64·37-s − 1.12·38-s + 0.810·40-s − 1.32·41-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: −1-1
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: 11
Selberg data: (2, 153, ( :3/2), −1)(2,\ 153,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
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 1−2.55T+8T2 1 - 2.55T + 8T^{2}
5 1+8.47T+125T2 1 + 8.47T + 125T^{2}
7 1−3.66T+343T2 1 - 3.66T + 343T^{2}
11 1+61.4T+1.33e3T2 1 + 61.4T + 1.33e3T^{2}
13 1−20.9T+2.19e3T2 1 - 20.9T + 2.19e3T^{2}
19 1+102.T+6.85e3T2 1 + 102.T + 6.85e3T^{2}
23 1−27.1T+1.21e4T2 1 - 27.1T + 1.21e4T^{2}
29 1−145.T+2.43e4T2 1 - 145.T + 2.43e4T^{2}
31 1−72.0T+2.97e4T2 1 - 72.0T + 2.97e4T^{2}
37 1−371.T+5.06e4T2 1 - 371.T + 5.06e4T^{2}
41 1+348.T+6.89e4T2 1 + 348.T + 6.89e4T^{2}
43 1+246.T+7.95e4T2 1 + 246.T + 7.95e4T^{2}
47 1−269.T+1.03e5T2 1 - 269.T + 1.03e5T^{2}
53 1−349.T+1.48e5T2 1 - 349.T + 1.48e5T^{2}
59 1+78.9T+2.05e5T2 1 + 78.9T + 2.05e5T^{2}
61 1+410.T+2.26e5T2 1 + 410.T + 2.26e5T^{2}
67 1+493.T+3.00e5T2 1 + 493.T + 3.00e5T^{2}
71 1+480.T+3.57e5T2 1 + 480.T + 3.57e5T^{2}
73 1−524.T+3.89e5T2 1 - 524.T + 3.89e5T^{2}
79 1+189.T+4.93e5T2 1 + 189.T + 4.93e5T^{2}
83 1−1.04e3T+5.71e5T2 1 - 1.04e3T + 5.71e5T^{2}
89 1+725.T+7.04e5T2 1 + 725.T + 7.04e5T^{2}
97 1+1.75e3T+9.12e5T2 1 + 1.75e3T + 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

−12.23544764524275146374043854873, −11.21163196919064666561215266294, −10.16420266429647573781039250093, −8.644243741732146534086964003420, −7.88229179255241623027027399994, −6.32754042281170777383415357055, −5.08004569547810011465154599049, −4.14971281380827279224524226351, −2.77838577401269894070988374506, 0, 2.77838577401269894070988374506, 4.14971281380827279224524226351, 5.08004569547810011465154599049, 6.32754042281170777383415357055, 7.88229179255241623027027399994, 8.644243741732146534086964003420, 10.16420266429647573781039250093, 11.21163196919064666561215266294, 12.23544764524275146374043854873

Graph of the ZZ-function along the critical line