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Properties

Label 2-57e2-1.1-c1-0-49
Degree 22
Conductor 32493249
Sign 11
Analytic cond. 25.943325.9433
Root an. cond. 5.093465.09346
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  + 0.741·2-s − 1.44·4-s + 3.30·5-s + 1.44·7-s − 2.55·8-s + 2.44·10-s − 1.81·11-s + 13-s + 1.07·14-s + 1.00·16-s − 6.60·17-s − 4.78·20-s − 1.34·22-s + 4.78·23-s + 5.89·25-s + 0.741·26-s − 2.10·28-s + 9.57·29-s + 4.55·31-s + 5.86·32-s − 4.89·34-s + 4.78·35-s + 5.89·37-s − 8.44·40-s + 2.96·41-s − 8.34·43-s + 2.63·44-s + ⋯
L(s)  = 1  + 0.524·2-s − 0.724·4-s + 1.47·5-s + 0.547·7-s − 0.904·8-s + 0.774·10-s − 0.547·11-s + 0.277·13-s + 0.287·14-s + 0.250·16-s − 1.60·17-s − 1.07·20-s − 0.287·22-s + 0.997·23-s + 1.17·25-s + 0.145·26-s − 0.397·28-s + 1.77·29-s + 0.817·31-s + 1.03·32-s − 0.840·34-s + 0.808·35-s + 0.969·37-s − 1.33·40-s + 0.463·41-s − 1.27·43-s + 0.397·44-s + ⋯

Functional equation

Λ(s)=(3249s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3249s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32493249    =    32⋅1923^{2} \cdot 19^{2}
Sign: 11
Analytic conductor: 25.943325.9433
Root analytic conductor: 5.093465.09346
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3249, ( :1/2), 1)(2,\ 3249,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.6878401462.687840146
L(12)L(\frac12) ≈\approx 2.6878401462.687840146
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
19 1 1
good2 1−0.741T+2T2 1 - 0.741T + 2T^{2}
5 1−3.30T+5T2 1 - 3.30T + 5T^{2}
7 1−1.44T+7T2 1 - 1.44T + 7T^{2}
11 1+1.81T+11T2 1 + 1.81T + 11T^{2}
13 1−T+13T2 1 - T + 13T^{2}
17 1+6.60T+17T2 1 + 6.60T + 17T^{2}
23 1−4.78T+23T2 1 - 4.78T + 23T^{2}
29 1−9.57T+29T2 1 - 9.57T + 29T^{2}
31 1−4.55T+31T2 1 - 4.55T + 31T^{2}
37 1−5.89T+37T2 1 - 5.89T + 37T^{2}
41 1−2.96T+41T2 1 - 2.96T + 41T^{2}
43 1+8.34T+43T2 1 + 8.34T + 43T^{2}
47 1−2.96T+47T2 1 - 2.96T + 47T^{2}
53 1−3.30T+53T2 1 - 3.30T + 53T^{2}
59 1−8.42T+59T2 1 - 8.42T + 59T^{2}
61 1−5T+61T2 1 - 5T + 61T^{2}
67 1−14.3T+67T2 1 - 14.3T + 67T^{2}
71 1+9.57T+71T2 1 + 9.57T + 71T^{2}
73 1−5T+73T2 1 - 5T + 73T^{2}
79 1−14.3T+79T2 1 - 14.3T + 79T^{2}
83 1−3.63T+83T2 1 - 3.63T + 83T^{2}
89 1+16.5T+89T2 1 + 16.5T + 89T^{2}
97 1+12.8T+97T2 1 + 12.8T + 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

−8.580118951534245605120114277661, −8.254998766042186178855840982241, −6.80194908973211238169930929909, −6.32826718444953341049294798766, −5.41888198969393245394789010153, −4.90453767821680727371247238748, −4.23739712688802755139141342699, −2.92004399062870730480409863400, −2.23486682153087132246635379017, −0.941935956128107851660332724334, 0.941935956128107851660332724334, 2.23486682153087132246635379017, 2.92004399062870730480409863400, 4.23739712688802755139141342699, 4.90453767821680727371247238748, 5.41888198969393245394789010153, 6.32826718444953341049294798766, 6.80194908973211238169930929909, 8.254998766042186178855840982241, 8.580118951534245605120114277661

Graph of the ZZ-function along the critical line