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Properties

Label 2-84-1.1-c3-0-0
Degree 22
Conductor 8484
Sign 11
Analytic cond. 4.956164.95616
Root an. cond. 2.226242.22624
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 6·5-s + 7·7-s + 9·9-s + 36·11-s + 62·13-s − 18·15-s + 114·17-s − 76·19-s − 21·21-s − 24·23-s − 89·25-s − 27·27-s + 54·29-s − 112·31-s − 108·33-s + 42·35-s − 178·37-s − 186·39-s + 378·41-s − 172·43-s + 54·45-s − 192·47-s + 49·49-s − 342·51-s − 402·53-s + 216·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.536·5-s + 0.377·7-s + 1/3·9-s + 0.986·11-s + 1.32·13-s − 0.309·15-s + 1.62·17-s − 0.917·19-s − 0.218·21-s − 0.217·23-s − 0.711·25-s − 0.192·27-s + 0.345·29-s − 0.648·31-s − 0.569·33-s + 0.202·35-s − 0.790·37-s − 0.763·39-s + 1.43·41-s − 0.609·43-s + 0.178·45-s − 0.595·47-s + 1/7·49-s − 0.939·51-s − 1.04·53-s + 0.529·55-s + ⋯

Functional equation

Λ(s)=(84s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(84s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8484    =    22⋅3⋅72^{2} \cdot 3 \cdot 7
Sign: 11
Analytic conductor: 4.956164.95616
Root analytic conductor: 2.226242.22624
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 84, ( :3/2), 1)(2,\ 84,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.5274694261.527469426
L(12)L(\frac12) ≈\approx 1.5274694261.527469426
L(52)L(\frac{5}{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)
bad2 1 1
3 1+pT 1 + p T
7 1−pT 1 - p T
good5 1−6T+p3T2 1 - 6 T + p^{3} T^{2}
11 1−36T+p3T2 1 - 36 T + p^{3} T^{2}
13 1−62T+p3T2 1 - 62 T + p^{3} T^{2}
17 1−114T+p3T2 1 - 114 T + p^{3} T^{2}
19 1+4pT+p3T2 1 + 4 p T + p^{3} T^{2}
23 1+24T+p3T2 1 + 24 T + p^{3} T^{2}
29 1−54T+p3T2 1 - 54 T + p^{3} T^{2}
31 1+112T+p3T2 1 + 112 T + p^{3} T^{2}
37 1+178T+p3T2 1 + 178 T + p^{3} T^{2}
41 1−378T+p3T2 1 - 378 T + p^{3} T^{2}
43 1+4pT+p3T2 1 + 4 p T + p^{3} T^{2}
47 1+192T+p3T2 1 + 192 T + p^{3} T^{2}
53 1+402T+p3T2 1 + 402 T + p^{3} T^{2}
59 1−396T+p3T2 1 - 396 T + p^{3} T^{2}
61 1−254T+p3T2 1 - 254 T + p^{3} T^{2}
67 1+1012T+p3T2 1 + 1012 T + p^{3} T^{2}
71 1−840T+p3T2 1 - 840 T + p^{3} T^{2}
73 1−890T+p3T2 1 - 890 T + p^{3} T^{2}
79 1−80T+p3T2 1 - 80 T + p^{3} T^{2}
83 1+108T+p3T2 1 + 108 T + p^{3} T^{2}
89 1+1638T+p3T2 1 + 1638 T + p^{3} T^{2}
97 1−1010T+p3T2 1 - 1010 T + p^{3} T^{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

−13.83201726133467512318490119763, −12.61856591791949487538252939790, −11.58908163730096665968715601934, −10.58315257361744133740154201358, −9.426228672939652439237526663022, −8.114389789415304142426981918414, −6.51779561773031791860008209629, −5.56030199737424878272255152376, −3.86728510554865626200271519258, −1.42982332922689640035017855530, 1.42982332922689640035017855530, 3.86728510554865626200271519258, 5.56030199737424878272255152376, 6.51779561773031791860008209629, 8.114389789415304142426981918414, 9.426228672939652439237526663022, 10.58315257361744133740154201358, 11.58908163730096665968715601934, 12.61856591791949487538252939790, 13.83201726133467512318490119763

Graph of the ZZ-function along the critical line