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Properties

Label 2-175-1.1-c3-0-9
Degree 22
Conductor 175175
Sign −1-1
Analytic cond. 10.325310.3253
Root an. cond. 3.213303.21330
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  − 2.85·2-s − 8.98·3-s + 0.149·4-s + 25.6·6-s − 7·7-s + 22.4·8-s + 53.7·9-s + 37.4·11-s − 1.34·12-s − 3.96·13-s + 19.9·14-s − 65.1·16-s − 51.6·17-s − 153.·18-s + 25.9·19-s + 62.9·21-s − 106.·22-s + 173.·23-s − 201.·24-s + 11.3·26-s − 240.·27-s − 1.04·28-s − 245.·29-s − 172.·31-s + 6.76·32-s − 336.·33-s + 147.·34-s + ⋯
L(s)  = 1  − 1.00·2-s − 1.72·3-s + 0.0186·4-s + 1.74·6-s − 0.377·7-s + 0.990·8-s + 1.99·9-s + 1.02·11-s − 0.0323·12-s − 0.0845·13-s + 0.381·14-s − 1.01·16-s − 0.737·17-s − 2.01·18-s + 0.313·19-s + 0.653·21-s − 1.03·22-s + 1.57·23-s − 1.71·24-s + 0.0853·26-s − 1.71·27-s − 0.00706·28-s − 1.57·29-s − 0.996·31-s + 0.0373·32-s − 1.77·33-s + 0.744·34-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: −1-1
Analytic conductor: 10.325310.3253
Root analytic conductor: 3.213303.21330
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 175, ( :3/2), −1)(2,\ 175,\ (\ :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)
bad5 1 1
7 1+7T 1 + 7T
good2 1+2.85T+8T2 1 + 2.85T + 8T^{2}
3 1+8.98T+27T2 1 + 8.98T + 27T^{2}
11 1−37.4T+1.33e3T2 1 - 37.4T + 1.33e3T^{2}
13 1+3.96T+2.19e3T2 1 + 3.96T + 2.19e3T^{2}
17 1+51.6T+4.91e3T2 1 + 51.6T + 4.91e3T^{2}
19 1−25.9T+6.85e3T2 1 - 25.9T + 6.85e3T^{2}
23 1−173.T+1.21e4T2 1 - 173.T + 1.21e4T^{2}
29 1+245.T+2.43e4T2 1 + 245.T + 2.43e4T^{2}
31 1+172.T+2.97e4T2 1 + 172.T + 2.97e4T^{2}
37 1−250.T+5.06e4T2 1 - 250.T + 5.06e4T^{2}
41 1+48.8T+6.89e4T2 1 + 48.8T + 6.89e4T^{2}
43 1−143.T+7.95e4T2 1 - 143.T + 7.95e4T^{2}
47 1−36.6T+1.03e5T2 1 - 36.6T + 1.03e5T^{2}
53 1+645.T+1.48e5T2 1 + 645.T + 1.48e5T^{2}
59 1−395.T+2.05e5T2 1 - 395.T + 2.05e5T^{2}
61 1−47.5T+2.26e5T2 1 - 47.5T + 2.26e5T^{2}
67 1+263.T+3.00e5T2 1 + 263.T + 3.00e5T^{2}
71 1+268.T+3.57e5T2 1 + 268.T + 3.57e5T^{2}
73 1+199.T+3.89e5T2 1 + 199.T + 3.89e5T^{2}
79 1−473.T+4.93e5T2 1 - 473.T + 4.93e5T^{2}
83 1−72.7T+5.71e5T2 1 - 72.7T + 5.71e5T^{2}
89 1+1.55e3T+7.04e5T2 1 + 1.55e3T + 7.04e5T^{2}
97 1+243.T+9.12e5T2 1 + 243.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

−11.28125826546917506493350277611, −10.96292159390626253631918404149, −9.701227383616021124175985670372, −9.060989595166397437626538911249, −7.39485695139970196937838556210, −6.58638271756813465217255796422, −5.35691336083601992329138299271, −4.18486505924493661207891435071, −1.28562818361079764931972099703, 0, 1.28562818361079764931972099703, 4.18486505924493661207891435071, 5.35691336083601992329138299271, 6.58638271756813465217255796422, 7.39485695139970196937838556210, 9.060989595166397437626538911249, 9.701227383616021124175985670372, 10.96292159390626253631918404149, 11.28125826546917506493350277611

Graph of the ZZ-function along the critical line