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Properties

Label 2-4205-1.1-c1-0-24
Degree 22
Conductor 42054205
Sign 11
Analytic cond. 33.577033.5770
Root an. cond. 5.794575.79457
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  + 3-s − 2·4-s − 5-s − 4.23·7-s − 2·9-s + 1.85·11-s − 2·12-s + 3.23·13-s − 15-s + 4·16-s − 1.85·17-s − 6.47·19-s + 2·20-s − 4.23·21-s − 7.85·23-s + 25-s − 5·27-s + 8.47·28-s + 5.47·31-s + 1.85·33-s + 4.23·35-s + 4·36-s − 5.76·37-s + 3.23·39-s − 9.70·41-s + 3.61·43-s − 3.70·44-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s − 0.447·5-s − 1.60·7-s − 0.666·9-s + 0.559·11-s − 0.577·12-s + 0.897·13-s − 0.258·15-s + 16-s − 0.449·17-s − 1.48·19-s + 0.447·20-s − 0.924·21-s − 1.63·23-s + 0.200·25-s − 0.962·27-s + 1.60·28-s + 0.982·31-s + 0.322·33-s + 0.716·35-s + 0.666·36-s − 0.947·37-s + 0.518·39-s − 1.51·41-s + 0.551·43-s − 0.559·44-s + ⋯

Functional equation

Λ(s)=(4205s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4205s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 42054205    =    5⋅2925 \cdot 29^{2}
Sign: 11
Analytic conductor: 33.577033.5770
Root analytic conductor: 5.794575.79457
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4205, ( :1/2), 1)(2,\ 4205,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.61396486460.6139648646
L(12)L(\frac12) ≈\approx 0.61396486460.6139648646
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)
bad5 1+T 1 + T
29 1 1
good2 1+2T2 1 + 2T^{2}
3 1−T+3T2 1 - T + 3T^{2}
7 1+4.23T+7T2 1 + 4.23T + 7T^{2}
11 1−1.85T+11T2 1 - 1.85T + 11T^{2}
13 1−3.23T+13T2 1 - 3.23T + 13T^{2}
17 1+1.85T+17T2 1 + 1.85T + 17T^{2}
19 1+6.47T+19T2 1 + 6.47T + 19T^{2}
23 1+7.85T+23T2 1 + 7.85T + 23T^{2}
31 1−5.47T+31T2 1 - 5.47T + 31T^{2}
37 1+5.76T+37T2 1 + 5.76T + 37T^{2}
41 1+9.70T+41T2 1 + 9.70T + 41T^{2}
43 1−3.61T+43T2 1 - 3.61T + 43T^{2}
47 1+9T+47T2 1 + 9T + 47T^{2}
53 1−3T+53T2 1 - 3T + 53T^{2}
59 1+6.70T+59T2 1 + 6.70T + 59T^{2}
61 1−10.3T+61T2 1 - 10.3T + 61T^{2}
67 1−12.2T+67T2 1 - 12.2T + 67T^{2}
71 1−8.56T+71T2 1 - 8.56T + 71T^{2}
73 1+2.32T+73T2 1 + 2.32T + 73T^{2}
79 1−12.2T+79T2 1 - 12.2T + 79T^{2}
83 1−2.29T+83T2 1 - 2.29T + 83T^{2}
89 1−6.70T+89T2 1 - 6.70T + 89T^{2}
97 1+1.94T+97T2 1 + 1.94T + 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

−8.331127389313440577377472207695, −8.175868242690917280480046892129, −6.69280298464385113273161308301, −6.36818524519496707801981969281, −5.51197108514426980058893554778, −4.33042201889232601851480557248, −3.71084203234995699264682797258, −3.27439670579290600344577135344, −2.08289937260327230825684404471, −0.41242305274089011301832990749, 0.41242305274089011301832990749, 2.08289937260327230825684404471, 3.27439670579290600344577135344, 3.71084203234995699264682797258, 4.33042201889232601851480557248, 5.51197108514426980058893554778, 6.36818524519496707801981969281, 6.69280298464385113273161308301, 8.175868242690917280480046892129, 8.331127389313440577377472207695

Graph of the ZZ-function along the critical line