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Properties

Label 2-1875-1.1-c1-0-44
Degree 22
Conductor 18751875
Sign −1-1
Analytic cond. 14.971914.9719
Root an. cond. 3.869363.86936
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.01·2-s − 3-s + 2.07·4-s + 2.01·6-s + 1.01·7-s − 0.153·8-s + 9-s + 4.75·11-s − 2.07·12-s − 0.103·13-s − 2.05·14-s − 3.84·16-s − 5.83·17-s − 2.01·18-s − 0.724·19-s − 1.01·21-s − 9.60·22-s − 9.07·23-s + 0.153·24-s + 0.209·26-s − 27-s + 2.11·28-s − 3.98·29-s + 1.06·31-s + 8.06·32-s − 4.75·33-s + 11.7·34-s + ⋯
L(s)  = 1  − 1.42·2-s − 0.577·3-s + 1.03·4-s + 0.824·6-s + 0.385·7-s − 0.0541·8-s + 0.333·9-s + 1.43·11-s − 0.599·12-s − 0.0287·13-s − 0.549·14-s − 0.960·16-s − 1.41·17-s − 0.475·18-s − 0.166·19-s − 0.222·21-s − 2.04·22-s − 1.89·23-s + 0.0312·24-s + 0.0411·26-s − 0.192·27-s + 0.399·28-s − 0.740·29-s + 0.191·31-s + 1.42·32-s − 0.828·33-s + 2.02·34-s + ⋯

Functional equation

Λ(s)=(1875s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(1875s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18751875    =    3⋅543 \cdot 5^{4}
Sign: −1-1
Analytic conductor: 14.971914.9719
Root analytic conductor: 3.869363.86936
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1875, ( :1/2), −1)(2,\ 1875,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{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+T 1 + T
5 1 1
good2 1+2.01T+2T2 1 + 2.01T + 2T^{2}
7 1−1.01T+7T2 1 - 1.01T + 7T^{2}
11 1−4.75T+11T2 1 - 4.75T + 11T^{2}
13 1+0.103T+13T2 1 + 0.103T + 13T^{2}
17 1+5.83T+17T2 1 + 5.83T + 17T^{2}
19 1+0.724T+19T2 1 + 0.724T + 19T^{2}
23 1+9.07T+23T2 1 + 9.07T + 23T^{2}
29 1+3.98T+29T2 1 + 3.98T + 29T^{2}
31 1−1.06T+31T2 1 - 1.06T + 31T^{2}
37 1−4.02T+37T2 1 - 4.02T + 37T^{2}
41 1−7.20T+41T2 1 - 7.20T + 41T^{2}
43 1−8.62T+43T2 1 - 8.62T + 43T^{2}
47 1+8.19T+47T2 1 + 8.19T + 47T^{2}
53 1+4.36T+53T2 1 + 4.36T + 53T^{2}
59 1+4.91T+59T2 1 + 4.91T + 59T^{2}
61 1−6.96T+61T2 1 - 6.96T + 61T^{2}
67 1+9.91T+67T2 1 + 9.91T + 67T^{2}
71 1+10.7T+71T2 1 + 10.7T + 71T^{2}
73 1−8.63T+73T2 1 - 8.63T + 73T^{2}
79 1−2.48T+79T2 1 - 2.48T + 79T^{2}
83 1−4.24T+83T2 1 - 4.24T + 83T^{2}
89 1+18.3T+89T2 1 + 18.3T + 89T^{2}
97 1−6.69T+97T2 1 - 6.69T + 97T^{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

−8.994526277468455021398467826058, −8.137902315445695142568240574491, −7.45799215653662537188243434312, −6.53797655296275849526972161110, −6.03720706488065652230960942451, −4.58986776968159248544434546122, −4.02029631003388117466887923934, −2.20779470816680977735835228784, −1.35661194388039420385169106014, 0, 1.35661194388039420385169106014, 2.20779470816680977735835228784, 4.02029631003388117466887923934, 4.58986776968159248544434546122, 6.03720706488065652230960942451, 6.53797655296275849526972161110, 7.45799215653662537188243434312, 8.137902315445695142568240574491, 8.994526277468455021398467826058

Graph of the ZZ-function along the critical line