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Properties

Label 2-1035-1.1-c1-0-32
Degree 22
Conductor 10351035
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.61·2-s + 4.85·4-s + 5-s + 1.23·7-s + 7.47·8-s + 2.61·10-s + 3.23·11-s − 6.23·13-s + 3.23·14-s + 9.85·16-s − 2.47·17-s − 5.70·19-s + 4.85·20-s + 8.47·22-s + 23-s + 25-s − 16.3·26-s + 6.00·28-s + 0.527·29-s + 4.23·31-s + 10.8·32-s − 6.47·34-s + 1.23·35-s − 9.70·37-s − 14.9·38-s + 7.47·40-s + 7.47·41-s + ⋯
L(s)  = 1  + 1.85·2-s + 2.42·4-s + 0.447·5-s + 0.467·7-s + 2.64·8-s + 0.827·10-s + 0.975·11-s − 1.72·13-s + 0.864·14-s + 2.46·16-s − 0.599·17-s − 1.30·19-s + 1.08·20-s + 1.80·22-s + 0.208·23-s + 0.200·25-s − 3.20·26-s + 1.13·28-s + 0.0980·29-s + 0.760·31-s + 1.91·32-s − 1.10·34-s + 0.208·35-s − 1.59·37-s − 2.42·38-s + 1.18·40-s + 1.16·41-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: 11
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: 00
Selberg data: (2, 1035, ( :1/2), 1)(2,\ 1035,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 5.3032288375.303228837
L(12)L(\frac12) ≈\approx 5.3032288375.303228837
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 1−2.61T+2T2 1 - 2.61T + 2T^{2}
7 1−1.23T+7T2 1 - 1.23T + 7T^{2}
11 1−3.23T+11T2 1 - 3.23T + 11T^{2}
13 1+6.23T+13T2 1 + 6.23T + 13T^{2}
17 1+2.47T+17T2 1 + 2.47T + 17T^{2}
19 1+5.70T+19T2 1 + 5.70T + 19T^{2}
29 1−0.527T+29T2 1 - 0.527T + 29T^{2}
31 1−4.23T+31T2 1 - 4.23T + 31T^{2}
37 1+9.70T+37T2 1 + 9.70T + 37T^{2}
41 1−7.47T+41T2 1 - 7.47T + 41T^{2}
43 1−3.70T+43T2 1 - 3.70T + 43T^{2}
47 1+9.47T+47T2 1 + 9.47T + 47T^{2}
53 1−6T+53T2 1 - 6T + 53T^{2}
59 1+8.94T+59T2 1 + 8.94T + 59T^{2}
61 1−12.1T+61T2 1 - 12.1T + 61T^{2}
67 1−9.70T+67T2 1 - 9.70T + 67T^{2}
71 1−6.23T+71T2 1 - 6.23T + 71T^{2}
73 1+6.70T+73T2 1 + 6.70T + 73T^{2}
79 1−8.76T+79T2 1 - 8.76T + 79T^{2}
83 1−6.47T+83T2 1 - 6.47T + 83T^{2}
89 1+2.76T+89T2 1 + 2.76T + 89T^{2}
97 1+6.18T+97T2 1 + 6.18T + 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

−10.22230213966789659233711101689, −9.193635076160769045052030264163, −7.990866990432419619198521853498, −6.84992571650857851397512948085, −6.51178108616119320555298530601, −5.35327197858425933061173837693, −4.70127714638929670600053208422, −3.95826615543655299208560770771, −2.66088764077004621332134736837, −1.87973744359409733703097371343, 1.87973744359409733703097371343, 2.66088764077004621332134736837, 3.95826615543655299208560770771, 4.70127714638929670600053208422, 5.35327197858425933061173837693, 6.51178108616119320555298530601, 6.84992571650857851397512948085, 7.990866990432419619198521853498, 9.193635076160769045052030264163, 10.22230213966789659233711101689

Graph of the ZZ-function along the critical line