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Properties

Label 2-175-1.1-c3-0-26
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  + 4.04·2-s − 6.52·3-s + 8.39·4-s − 26.4·6-s − 7·7-s + 1.58·8-s + 15.6·9-s + 6.78·11-s − 54.7·12-s − 48.9·13-s − 28.3·14-s − 60.7·16-s − 92.4·17-s + 63.1·18-s − 125.·19-s + 45.6·21-s + 27.4·22-s + 32.2·23-s − 10.3·24-s − 198.·26-s + 74.3·27-s − 58.7·28-s + 282.·29-s + 205.·31-s − 258.·32-s − 44.2·33-s − 374.·34-s + ⋯
L(s)  = 1  + 1.43·2-s − 1.25·3-s + 1.04·4-s − 1.79·6-s − 0.377·7-s + 0.0698·8-s + 0.578·9-s + 0.185·11-s − 1.31·12-s − 1.04·13-s − 0.541·14-s − 0.948·16-s − 1.31·17-s + 0.827·18-s − 1.51·19-s + 0.474·21-s + 0.266·22-s + 0.292·23-s − 0.0877·24-s − 1.49·26-s + 0.530·27-s − 0.396·28-s + 1.81·29-s + 1.19·31-s − 1.42·32-s − 0.233·33-s − 1.88·34-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 1−4.04T+8T2 1 - 4.04T + 8T^{2}
3 1+6.52T+27T2 1 + 6.52T + 27T^{2}
11 1−6.78T+1.33e3T2 1 - 6.78T + 1.33e3T^{2}
13 1+48.9T+2.19e3T2 1 + 48.9T + 2.19e3T^{2}
17 1+92.4T+4.91e3T2 1 + 92.4T + 4.91e3T^{2}
19 1+125.T+6.85e3T2 1 + 125.T + 6.85e3T^{2}
23 1−32.2T+1.21e4T2 1 - 32.2T + 1.21e4T^{2}
29 1−282.T+2.43e4T2 1 - 282.T + 2.43e4T^{2}
31 1−205.T+2.97e4T2 1 - 205.T + 2.97e4T^{2}
37 1+190.T+5.06e4T2 1 + 190.T + 5.06e4T^{2}
41 1−123.T+6.89e4T2 1 - 123.T + 6.89e4T^{2}
43 1+35.0T+7.95e4T2 1 + 35.0T + 7.95e4T^{2}
47 1+419.T+1.03e5T2 1 + 419.T + 1.03e5T^{2}
53 1−0.365T+1.48e5T2 1 - 0.365T + 1.48e5T^{2}
59 1−328.T+2.05e5T2 1 - 328.T + 2.05e5T^{2}
61 1+515.T+2.26e5T2 1 + 515.T + 2.26e5T^{2}
67 1−828.T+3.00e5T2 1 - 828.T + 3.00e5T^{2}
71 1+496.T+3.57e5T2 1 + 496.T + 3.57e5T^{2}
73 1+701.T+3.89e5T2 1 + 701.T + 3.89e5T^{2}
79 1−199.T+4.93e5T2 1 - 199.T + 4.93e5T^{2}
83 1−194.T+5.71e5T2 1 - 194.T + 5.71e5T^{2}
89 1−137.T+7.04e5T2 1 - 137.T + 7.04e5T^{2}
97 1+220.T+9.12e5T2 1 + 220.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

−12.03800311785481645089385689920, −11.16346308153900662304801112102, −10.18794925764385988798726158836, −8.692080342934056033971275683084, −6.70309320644208085946320962263, −6.34139052150607437890645164701, −5.01780550110484013517578075560, −4.37161285307502519328235807274, −2.63054893822071619729802472483, 0, 2.63054893822071619729802472483, 4.37161285307502519328235807274, 5.01780550110484013517578075560, 6.34139052150607437890645164701, 6.70309320644208085946320962263, 8.692080342934056033971275683084, 10.18794925764385988798726158836, 11.16346308153900662304801112102, 12.03800311785481645089385689920

Graph of the ZZ-function along the critical line