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Properties

Label 2-1035-1.1-c1-0-18
Degree 22
Conductor 10351035
Sign −1-1
Analytic cond. 8.264518.26451
Root an. cond. 2.874802.87480
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  − 2.69·2-s + 5.25·4-s − 5-s − 2.74·7-s − 8.76·8-s + 2.69·10-s + 3.38·11-s − 2.46·13-s + 7.38·14-s + 13.1·16-s + 2.64·17-s + 3.38·19-s − 5.25·20-s − 9.12·22-s + 23-s + 25-s + 6.63·26-s − 14.4·28-s − 9.20·29-s − 5.10·31-s − 17.7·32-s − 7.12·34-s + 2.74·35-s + 5.50·37-s − 9.12·38-s + 8.76·40-s + 1.20·41-s + ⋯
L(s)  = 1  − 1.90·2-s + 2.62·4-s − 0.447·5-s − 1.03·7-s − 3.09·8-s + 0.851·10-s + 1.02·11-s − 0.683·13-s + 1.97·14-s + 3.27·16-s + 0.641·17-s + 0.777·19-s − 1.17·20-s − 1.94·22-s + 0.208·23-s + 0.200·25-s + 1.30·26-s − 2.72·28-s − 1.70·29-s − 0.917·31-s − 3.14·32-s − 1.22·34-s + 0.463·35-s + 0.904·37-s − 1.47·38-s + 1.38·40-s + 0.188·41-s + ⋯

Functional equation

Λ(s)=(1035s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(1035s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10351035    =    32⋅5⋅233^{2} \cdot 5 \cdot 23
Sign: −1-1
Analytic conductor: 8.264518.26451
Root analytic conductor: 2.874802.87480
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1035, ( :1/2), −1)(2,\ 1035,\ (\ :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 1
5 1+T 1 + T
23 1−T 1 - T
good2 1+2.69T+2T2 1 + 2.69T + 2T^{2}
7 1+2.74T+7T2 1 + 2.74T + 7T^{2}
11 1−3.38T+11T2 1 - 3.38T + 11T^{2}
13 1+2.46T+13T2 1 + 2.46T + 13T^{2}
17 1−2.64T+17T2 1 - 2.64T + 17T^{2}
19 1−3.38T+19T2 1 - 3.38T + 19T^{2}
29 1+9.20T+29T2 1 + 9.20T + 29T^{2}
31 1+5.10T+31T2 1 + 5.10T + 31T^{2}
37 1−5.50T+37T2 1 - 5.50T + 37T^{2}
41 1−1.20T+41T2 1 - 1.20T + 41T^{2}
43 1−3.02T+43T2 1 - 3.02T + 43T^{2}
47 1+8.21T+47T2 1 + 8.21T + 47T^{2}
53 1−10.5T+53T2 1 - 10.5T + 53T^{2}
59 1+8.28T+59T2 1 + 8.28T + 59T^{2}
61 1−0.263T+61T2 1 - 0.263T + 61T^{2}
67 1+7.66T+67T2 1 + 7.66T + 67T^{2}
71 1−0.0150T+71T2 1 - 0.0150T + 71T^{2}
73 1+5.53T+73T2 1 + 5.53T + 73T^{2}
79 1+8.67T+79T2 1 + 8.67T + 79T^{2}
83 1−3.52T+83T2 1 - 3.52T + 83T^{2}
89 1−4.66T+89T2 1 - 4.66T + 89T^{2}
97 1+4.09T+97T2 1 + 4.09T + 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

−9.407652145642897221537261120838, −9.015998627137688328846387142727, −7.81506847329111350928777685572, −7.31340061162445015985951847296, −6.55764277109648950551944504831, −5.62958374639956538768473228668, −3.74879232965867172154814975152, −2.79708757196334696687636319371, −1.38864983402180821709110777698, 0, 1.38864983402180821709110777698, 2.79708757196334696687636319371, 3.74879232965867172154814975152, 5.62958374639956538768473228668, 6.55764277109648950551944504831, 7.31340061162445015985951847296, 7.81506847329111350928777685572, 9.015998627137688328846387142727, 9.407652145642897221537261120838

Graph of the ZZ-function along the critical line