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Properties

Label 2-175-1.1-c3-0-18
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  − 3.70·2-s + 5.70·3-s + 5.70·4-s − 21.1·6-s − 7·7-s + 8.50·8-s + 5.50·9-s − 60.0·11-s + 32.5·12-s − 0.387·13-s + 25.9·14-s − 77.1·16-s + 35.4·17-s − 20.3·18-s − 6.08·19-s − 39.9·21-s + 222.·22-s + 31.5·23-s + 48.5·24-s + 1.43·26-s − 122.·27-s − 39.9·28-s − 292.·29-s + 130.·31-s + 217.·32-s − 342.·33-s − 131.·34-s + ⋯
L(s)  = 1  − 1.30·2-s + 1.09·3-s + 0.712·4-s − 1.43·6-s − 0.377·7-s + 0.375·8-s + 0.203·9-s − 1.64·11-s + 0.782·12-s − 0.00826·13-s + 0.494·14-s − 1.20·16-s + 0.506·17-s − 0.266·18-s − 0.0735·19-s − 0.414·21-s + 2.15·22-s + 0.285·23-s + 0.412·24-s + 0.0108·26-s − 0.873·27-s − 0.269·28-s − 1.87·29-s + 0.754·31-s + 1.20·32-s − 1.80·33-s − 0.662·34-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+3.70T+8T2 1 + 3.70T + 8T^{2}
3 1−5.70T+27T2 1 - 5.70T + 27T^{2}
11 1+60.0T+1.33e3T2 1 + 60.0T + 1.33e3T^{2}
13 1+0.387T+2.19e3T2 1 + 0.387T + 2.19e3T^{2}
17 1−35.4T+4.91e3T2 1 - 35.4T + 4.91e3T^{2}
19 1+6.08T+6.85e3T2 1 + 6.08T + 6.85e3T^{2}
23 1−31.5T+1.21e4T2 1 - 31.5T + 1.21e4T^{2}
29 1+292.T+2.43e4T2 1 + 292.T + 2.43e4T^{2}
31 1−130.T+2.97e4T2 1 - 130.T + 2.97e4T^{2}
37 1+219.T+5.06e4T2 1 + 219.T + 5.06e4T^{2}
41 1+447.T+6.89e4T2 1 + 447.T + 6.89e4T^{2}
43 1+210.T+7.95e4T2 1 + 210.T + 7.95e4T^{2}
47 1+457.T+1.03e5T2 1 + 457.T + 1.03e5T^{2}
53 1+144.T+1.48e5T2 1 + 144.T + 1.48e5T^{2}
59 1−767.T+2.05e5T2 1 - 767.T + 2.05e5T^{2}
61 1−667.T+2.26e5T2 1 - 667.T + 2.26e5T^{2}
67 1−77.4T+3.00e5T2 1 - 77.4T + 3.00e5T^{2}
71 1+906.T+3.57e5T2 1 + 906.T + 3.57e5T^{2}
73 1−1.02e3T+3.89e5T2 1 - 1.02e3T + 3.89e5T^{2}
79 1+690.T+4.93e5T2 1 + 690.T + 4.93e5T^{2}
83 1−979.T+5.71e5T2 1 - 979.T + 5.71e5T^{2}
89 1+910.T+7.04e5T2 1 + 910.T + 7.04e5T^{2}
97 1+11.1T+9.12e5T2 1 + 11.1T + 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

−11.41832470961225980992717063400, −10.27225130164580296470587129842, −9.676777644196231857586231896516, −8.573381728798433632334930724805, −8.030839375156797704516647440047, −7.08053852838920900218055868615, −5.24844023302266390233237627885, −3.34707444407198510518869819137, −2.05284208090969814397299941670, 0, 2.05284208090969814397299941670, 3.34707444407198510518869819137, 5.24844023302266390233237627885, 7.08053852838920900218055868615, 8.030839375156797704516647440047, 8.573381728798433632334930724805, 9.676777644196231857586231896516, 10.27225130164580296470587129842, 11.41832470961225980992717063400

Graph of the ZZ-function along the critical line