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Properties

Label 2-675-1.1-c3-0-37
Degree 22
Conductor 675675
Sign −1-1
Analytic cond. 39.826239.8262
Root an. cond. 6.310806.31080
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  − 3.60·2-s + 4.99·4-s − 9·7-s + 10.8·8-s − 14.4·11-s + 34·13-s + 32.4·14-s − 79·16-s + 7.21·17-s − 101·19-s + 51.9·22-s + 108.·23-s − 122.·26-s − 44.9·28-s + 165.·29-s − 3·31-s + 198.·32-s − 25.9·34-s − 67·37-s + 364.·38-s − 201.·41-s − 137·43-s − 72.1·44-s − 390·46-s + 115.·47-s − 262·49-s + 169.·52-s + ⋯
L(s)  = 1  − 1.27·2-s + 0.624·4-s − 0.485·7-s + 0.478·8-s − 0.395·11-s + 0.725·13-s + 0.619·14-s − 1.23·16-s + 0.102·17-s − 1.21·19-s + 0.503·22-s + 0.980·23-s − 0.924·26-s − 0.303·28-s + 1.06·29-s − 0.0173·31-s + 1.09·32-s − 0.131·34-s − 0.297·37-s + 1.55·38-s − 0.769·41-s − 0.485·43-s − 0.247·44-s − 1.25·46-s + 0.358·47-s − 0.763·49-s + 0.453·52-s + ⋯

Functional equation

Λ(s)=(675s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(675s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 675675    =    33⋅523^{3} \cdot 5^{2}
Sign: −1-1
Analytic conductor: 39.826239.8262
Root analytic conductor: 6.310806.31080
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 675, ( :3/2), −1)(2,\ 675,\ (\ :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
5 1 1
good2 1+3.60T+8T2 1 + 3.60T + 8T^{2}
7 1+9T+343T2 1 + 9T + 343T^{2}
11 1+14.4T+1.33e3T2 1 + 14.4T + 1.33e3T^{2}
13 1−34T+2.19e3T2 1 - 34T + 2.19e3T^{2}
17 1−7.21T+4.91e3T2 1 - 7.21T + 4.91e3T^{2}
19 1+101T+6.85e3T2 1 + 101T + 6.85e3T^{2}
23 1−108.T+1.21e4T2 1 - 108.T + 1.21e4T^{2}
29 1−165.T+2.43e4T2 1 - 165.T + 2.43e4T^{2}
31 1+3T+2.97e4T2 1 + 3T + 2.97e4T^{2}
37 1+67T+5.06e4T2 1 + 67T + 5.06e4T^{2}
41 1+201.T+6.89e4T2 1 + 201.T + 6.89e4T^{2}
43 1+137T+7.95e4T2 1 + 137T + 7.95e4T^{2}
47 1−115.T+1.03e5T2 1 - 115.T + 1.03e5T^{2}
53 1−677.T+1.48e5T2 1 - 677.T + 1.48e5T^{2}
59 1−872.T+2.05e5T2 1 - 872.T + 2.05e5T^{2}
61 1+563T+2.26e5T2 1 + 563T + 2.26e5T^{2}
67 1−1.04e3T+3.00e5T2 1 - 1.04e3T + 3.00e5T^{2}
71 1+786.T+3.57e5T2 1 + 786.T + 3.57e5T^{2}
73 1+503T+3.89e5T2 1 + 503T + 3.89e5T^{2}
79 1−615T+4.93e5T2 1 - 615T + 4.93e5T^{2}
83 1+237.T+5.71e5T2 1 + 237.T + 5.71e5T^{2}
89 1+973.T+7.04e5T2 1 + 973.T + 7.04e5T^{2}
97 1+19T+9.12e5T2 1 + 19T + 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

−9.653615177128799339858546407925, −8.658644378013392503406004328059, −8.327029873538318416435291916222, −7.12933377947154105195811115838, −6.45137704185909663750377940568, −5.14299765028413885321342381311, −3.95009122690299283750359061872, −2.56876196431752541391331981678, −1.21980592044597267614797041905, 0, 1.21980592044597267614797041905, 2.56876196431752541391331981678, 3.95009122690299283750359061872, 5.14299765028413885321342381311, 6.45137704185909663750377940568, 7.12933377947154105195811115838, 8.327029873538318416435291916222, 8.658644378013392503406004328059, 9.653615177128799339858546407925

Graph of the ZZ-function along the critical line