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Properties

Label 2-294-1.1-c5-0-1
Degree 22
Conductor 294294
Sign 11
Analytic cond. 47.152847.1528
Root an. cond. 6.866796.86679
Motivic weight 55
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s − 9·3-s + 16·4-s − 26·5-s + 36·6-s − 64·8-s + 81·9-s + 104·10-s − 358·11-s − 144·12-s − 332·13-s + 234·15-s + 256·16-s − 126·17-s − 324·18-s + 2.20e3·19-s − 416·20-s + 1.43e3·22-s − 2.14e3·23-s + 576·24-s − 2.44e3·25-s + 1.32e3·26-s − 729·27-s − 3.61e3·29-s − 936·30-s − 5.66e3·31-s − 1.02e3·32-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.465·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.328·10-s − 0.892·11-s − 0.288·12-s − 0.544·13-s + 0.268·15-s + 1/4·16-s − 0.105·17-s − 0.235·18-s + 1.39·19-s − 0.232·20-s + 0.630·22-s − 0.844·23-s + 0.204·24-s − 0.783·25-s + 0.385·26-s − 0.192·27-s − 0.797·29-s − 0.189·30-s − 1.05·31-s − 0.176·32-s + ⋯

Functional equation

Λ(s)=(294s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(294s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 294294    =    2⋅3⋅722 \cdot 3 \cdot 7^{2}
Sign: 11
Analytic conductor: 47.152847.1528
Root analytic conductor: 6.866796.86679
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 294, ( :5/2), 1)(2,\ 294,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 0.60096471630.6009647163
L(12)L(\frac12) ≈\approx 0.60096471630.6009647163
L(72)L(\frac{7}{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+p2T 1 + p^{2} T
3 1+p2T 1 + p^{2} T
7 1 1
good5 1+26T+p5T2 1 + 26 T + p^{5} T^{2}
11 1+358T+p5T2 1 + 358 T + p^{5} T^{2}
13 1+332T+p5T2 1 + 332 T + p^{5} T^{2}
17 1+126T+p5T2 1 + 126 T + p^{5} T^{2}
19 1−2200T+p5T2 1 - 2200 T + p^{5} T^{2}
23 1+2142T+p5T2 1 + 2142 T + p^{5} T^{2}
29 1+3610T+p5T2 1 + 3610 T + p^{5} T^{2}
31 1+5668T+p5T2 1 + 5668 T + p^{5} T^{2}
37 1+2922T+p5T2 1 + 2922 T + p^{5} T^{2}
41 1−2142T+p5T2 1 - 2142 T + p^{5} T^{2}
43 1−6388T+p5T2 1 - 6388 T + p^{5} T^{2}
47 1−6520T+p5T2 1 - 6520 T + p^{5} T^{2}
53 1+10702T+p5T2 1 + 10702 T + p^{5} T^{2}
59 1+42524T+p5T2 1 + 42524 T + p^{5} T^{2}
61 1−44840T+p5T2 1 - 44840 T + p^{5} T^{2}
67 1+1448T+p5T2 1 + 1448 T + p^{5} T^{2}
71 1+62pT+p5T2 1 + 62 p T + p^{5} T^{2}
73 1+20500T+p5T2 1 + 20500 T + p^{5} T^{2}
79 1−65236T+p5T2 1 - 65236 T + p^{5} T^{2}
83 1−102804T+p5T2 1 - 102804 T + p^{5} T^{2}
89 1−128006T+p5T2 1 - 128006 T + p^{5} T^{2}
97 1−113324T+p5T2 1 - 113324 T + p^{5} 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

−10.89998889112775390931188380798, −10.01551586200578884775026638630, −9.177276495302374498963343358431, −7.77714911097757860906220121471, −7.41255817535828905180997149700, −5.99952106366225872595590896861, −5.03939370182261565538025126896, −3.55332159566161255377636934530, −2.05298204378893589144892360531, −0.48421193587887993428961725216, 0.48421193587887993428961725216, 2.05298204378893589144892360531, 3.55332159566161255377636934530, 5.03939370182261565538025126896, 5.99952106366225872595590896861, 7.41255817535828905180997149700, 7.77714911097757860906220121471, 9.177276495302374498963343358431, 10.01551586200578884775026638630, 10.89998889112775390931188380798

Graph of the ZZ-function along the critical line