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Properties

Label 2-175-1.1-c3-0-20
Degree 22
Conductor 175175
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.31·2-s + 1.93·3-s + 20.2·4-s + 10.3·6-s + 7·7-s + 65.0·8-s − 23.2·9-s + 25.5·11-s + 39.2·12-s − 64.1·13-s + 37.1·14-s + 183.·16-s − 27.6·17-s − 123.·18-s + 0.792·19-s + 13.5·21-s + 135.·22-s + 108.·23-s + 126.·24-s − 340.·26-s − 97.4·27-s + 141.·28-s − 234.·29-s + 129.·31-s + 455.·32-s + 49.5·33-s − 147.·34-s + ⋯
L(s)  = 1  + 1.87·2-s + 0.373·3-s + 2.52·4-s + 0.701·6-s + 0.377·7-s + 2.87·8-s − 0.860·9-s + 0.700·11-s + 0.944·12-s − 1.36·13-s + 0.710·14-s + 2.86·16-s − 0.395·17-s − 1.61·18-s + 0.00956·19-s + 0.141·21-s + 1.31·22-s + 0.984·23-s + 1.07·24-s − 2.56·26-s − 0.694·27-s + 0.956·28-s − 1.49·29-s + 0.748·31-s + 2.51·32-s + 0.261·33-s − 0.742·34-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: 11
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: 00
Selberg data: (2, 175, ( :3/2), 1)(2,\ 175,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 5.5582697225.558269722
L(12)L(\frac12) ≈\approx 5.5582697225.558269722
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 1−5.31T+8T2 1 - 5.31T + 8T^{2}
3 1−1.93T+27T2 1 - 1.93T + 27T^{2}
11 1−25.5T+1.33e3T2 1 - 25.5T + 1.33e3T^{2}
13 1+64.1T+2.19e3T2 1 + 64.1T + 2.19e3T^{2}
17 1+27.6T+4.91e3T2 1 + 27.6T + 4.91e3T^{2}
19 1−0.792T+6.85e3T2 1 - 0.792T + 6.85e3T^{2}
23 1−108.T+1.21e4T2 1 - 108.T + 1.21e4T^{2}
29 1+234.T+2.43e4T2 1 + 234.T + 2.43e4T^{2}
31 1−129.T+2.97e4T2 1 - 129.T + 2.97e4T^{2}
37 1−38.3T+5.06e4T2 1 - 38.3T + 5.06e4T^{2}
41 1+403.T+6.89e4T2 1 + 403.T + 6.89e4T^{2}
43 1−172.T+7.95e4T2 1 - 172.T + 7.95e4T^{2}
47 1−206.T+1.03e5T2 1 - 206.T + 1.03e5T^{2}
53 1−144.T+1.48e5T2 1 - 144.T + 1.48e5T^{2}
59 1+679.T+2.05e5T2 1 + 679.T + 2.05e5T^{2}
61 1+574.T+2.26e5T2 1 + 574.T + 2.26e5T^{2}
67 1−515.T+3.00e5T2 1 - 515.T + 3.00e5T^{2}
71 1−556.T+3.57e5T2 1 - 556.T + 3.57e5T^{2}
73 1+173.T+3.89e5T2 1 + 173.T + 3.89e5T^{2}
79 1−79.3T+4.93e5T2 1 - 79.3T + 4.93e5T^{2}
83 1−1.04e3T+5.71e5T2 1 - 1.04e3T + 5.71e5T^{2}
89 1−652.T+7.04e5T2 1 - 652.T + 7.04e5T^{2}
97 1−515.T+9.12e5T2 1 - 515.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

−12.33347982917192994879861594794, −11.65584461523976225293688474095, −10.80015034965693102238268039423, −9.233008820908659393501233777773, −7.72245743193950273728188066175, −6.70617159427420420140411328914, −5.51016825489828582941858973925, −4.57395809141052063824924243407, −3.29365934194548690976233143321, −2.15785001882933881199835507726, 2.15785001882933881199835507726, 3.29365934194548690976233143321, 4.57395809141052063824924243407, 5.51016825489828582941858973925, 6.70617159427420420140411328914, 7.72245743193950273728188066175, 9.233008820908659393501233777773, 10.80015034965693102238268039423, 11.65584461523976225293688474095, 12.33347982917192994879861594794

Graph of the ZZ-function along the critical line