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Properties

Label 2-546-1.1-c1-0-3
Degree 22
Conductor 546546
Sign 11
Analytic cond. 4.359834.35983
Root an. cond. 2.088022.08802
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-s − 3-s + 4-s + 3.27·5-s + 6-s + 7-s − 8-s + 9-s − 3.27·10-s + 5.27·11-s − 12-s − 13-s − 14-s − 3.27·15-s + 16-s − 7.27·17-s − 18-s + 5.27·19-s + 3.27·20-s − 21-s − 5.27·22-s − 5.27·23-s + 24-s + 5.72·25-s + 26-s − 27-s + 28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.46·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s − 1.03·10-s + 1.59·11-s − 0.288·12-s − 0.277·13-s − 0.267·14-s − 0.845·15-s + 0.250·16-s − 1.76·17-s − 0.235·18-s + 1.21·19-s + 0.732·20-s − 0.218·21-s − 1.12·22-s − 1.09·23-s + 0.204·24-s + 1.14·25-s + 0.196·26-s − 0.192·27-s + 0.188·28-s + ⋯

Functional equation

Λ(s)=(546s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(546s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 546546    =    2⋅3⋅7⋅132 \cdot 3 \cdot 7 \cdot 13
Sign: 11
Analytic conductor: 4.359834.35983
Root analytic conductor: 2.088022.08802
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 546, ( :1/2), 1)(2,\ 546,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.2376349021.237634902
L(12)L(\frac12) ≈\approx 1.2376349021.237634902
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)
bad2 1+T 1 + T
3 1+T 1 + T
7 1−T 1 - T
13 1+T 1 + T
good5 1−3.27T+5T2 1 - 3.27T + 5T^{2}
11 1−5.27T+11T2 1 - 5.27T + 11T^{2}
17 1+7.27T+17T2 1 + 7.27T + 17T^{2}
19 1−5.27T+19T2 1 - 5.27T + 19T^{2}
23 1+5.27T+23T2 1 + 5.27T + 23T^{2}
29 1−0.725T+29T2 1 - 0.725T + 29T^{2}
31 1−8T+31T2 1 - 8T + 31T^{2}
37 1−3.27T+37T2 1 - 3.27T + 37T^{2}
41 1+8.54T+41T2 1 + 8.54T + 41T^{2}
43 1−5.27T+43T2 1 - 5.27T + 43T^{2}
47 1+47T2 1 + 47T^{2}
53 1−10T+53T2 1 - 10T + 53T^{2}
59 1−8T+59T2 1 - 8T + 59T^{2}
61 1+4.72T+61T2 1 + 4.72T + 61T^{2}
67 1+2.54T+67T2 1 + 2.54T + 67T^{2}
71 1+71T2 1 + 71T^{2}
73 1−9.82T+73T2 1 - 9.82T + 73T^{2}
79 1+2.54T+79T2 1 + 2.54T + 79T^{2}
83 1−10.5T+83T2 1 - 10.5T + 83T^{2}
89 1+14T+89T2 1 + 14T + 89T^{2}
97 1+15.0T+97T2 1 + 15.0T + 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

−10.66146478055000204937840010278, −9.767772604161815803693236117002, −9.309884670475511043909031425770, −8.343518506082386869702031908164, −6.91026514947932260405007550027, −6.39870794343943751494465686215, −5.47193319690133672490048174894, −4.24062375293323472675955780464, −2.35289732891516897669516912027, −1.28606116306891868803211902621, 1.28606116306891868803211902621, 2.35289732891516897669516912027, 4.24062375293323472675955780464, 5.47193319690133672490048174894, 6.39870794343943751494465686215, 6.91026514947932260405007550027, 8.343518506082386869702031908164, 9.309884670475511043909031425770, 9.767772604161815803693236117002, 10.66146478055000204937840010278

Graph of the ZZ-function along the critical line