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Properties

Label 2-1035-1.1-c1-0-19
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  − 1.32·2-s − 0.231·4-s − 5-s − 3.50·7-s + 2.96·8-s + 1.32·10-s + 0.659·11-s + 5.91·13-s + 4.65·14-s − 3.48·16-s − 0.844·17-s + 0.659·19-s + 0.231·20-s − 0.876·22-s + 23-s + 25-s − 7.85·26-s + 0.812·28-s − 1.59·29-s + 6.75·31-s − 1.30·32-s + 1.12·34-s + 3.50·35-s − 11.7·37-s − 0.876·38-s − 2.96·40-s − 6.40·41-s + ⋯
L(s)  = 1  − 0.940·2-s − 0.115·4-s − 0.447·5-s − 1.32·7-s + 1.04·8-s + 0.420·10-s + 0.198·11-s + 1.63·13-s + 1.24·14-s − 0.870·16-s − 0.204·17-s + 0.151·19-s + 0.0518·20-s − 0.186·22-s + 0.208·23-s + 0.200·25-s − 1.54·26-s + 0.153·28-s − 0.295·29-s + 1.21·31-s − 0.230·32-s + 0.192·34-s + 0.592·35-s − 1.93·37-s − 0.142·38-s − 0.469·40-s − 1.00·41-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+1.32T+2T2 1 + 1.32T + 2T^{2}
7 1+3.50T+7T2 1 + 3.50T + 7T^{2}
11 1−0.659T+11T2 1 - 0.659T + 11T^{2}
13 1−5.91T+13T2 1 - 5.91T + 13T^{2}
17 1+0.844T+17T2 1 + 0.844T + 17T^{2}
19 1−0.659T+19T2 1 - 0.659T + 19T^{2}
29 1+1.59T+29T2 1 + 1.59T + 29T^{2}
31 1−6.75T+31T2 1 - 6.75T + 31T^{2}
37 1+11.7T+37T2 1 + 11.7T + 37T^{2}
41 1+6.40T+41T2 1 + 6.40T + 41T^{2}
43 1+9.47T+43T2 1 + 9.47T + 43T^{2}
47 1+6.88T+47T2 1 + 6.88T + 47T^{2}
53 1+6.64T+53T2 1 + 6.64T + 53T^{2}
59 1−4.97T+59T2 1 - 4.97T + 59T^{2}
61 1−5.78T+61T2 1 - 5.78T + 61T^{2}
67 1−8.31T+67T2 1 - 8.31T + 67T^{2}
71 1−3.63T+71T2 1 - 3.63T + 71T^{2}
73 1+13.9T+73T2 1 + 13.9T + 73T^{2}
79 1−1.02T+79T2 1 - 1.02T + 79T^{2}
83 1−8.27T+83T2 1 - 8.27T + 83T^{2}
89 1+17.6T+89T2 1 + 17.6T + 89T^{2}
97 1+8.34T+97T2 1 + 8.34T + 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

−9.536459176444697601927410166285, −8.566755606491236533339775438188, −8.290729113865955435769874305499, −6.95209444415982296854284717705, −6.48147426951550840825344547782, −5.19904895479398377103049729230, −3.94986416066646605210272588956, −3.23851115866038196686041551428, −1.41675639146642653919743358536, 0, 1.41675639146642653919743358536, 3.23851115866038196686041551428, 3.94986416066646605210272588956, 5.19904895479398377103049729230, 6.48147426951550840825344547782, 6.95209444415982296854284717705, 8.290729113865955435769874305499, 8.566755606491236533339775438188, 9.536459176444697601927410166285

Graph of the ZZ-function along the critical line