Properties

Label 2-51-1.1-c1-0-1
Degree $2$
Conductor $51$
Sign $1$
Analytic cond. $0.407237$
Root an. cond. $0.638151$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·4-s + 3·5-s − 4·7-s + 9-s − 3·11-s − 2·12-s − 13-s + 3·15-s + 4·16-s − 17-s − 19-s − 6·20-s − 4·21-s + 9·23-s + 4·25-s + 27-s + 8·28-s + 6·29-s + 2·31-s − 3·33-s − 12·35-s − 2·36-s − 4·37-s − 39-s − 3·41-s − 7·43-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s + 1.34·5-s − 1.51·7-s + 1/3·9-s − 0.904·11-s − 0.577·12-s − 0.277·13-s + 0.774·15-s + 16-s − 0.242·17-s − 0.229·19-s − 1.34·20-s − 0.872·21-s + 1.87·23-s + 4/5·25-s + 0.192·27-s + 1.51·28-s + 1.11·29-s + 0.359·31-s − 0.522·33-s − 2.02·35-s − 1/3·36-s − 0.657·37-s − 0.160·39-s − 0.468·41-s − 1.06·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(51\)    =    \(3 \cdot 17\)
Sign: $1$
Analytic conductor: \(0.407237\)
Root analytic conductor: \(0.638151\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 51,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8600590161\)
\(L(\frac12)\) \(\approx\) \(0.8600590161\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
17 \( 1 + T \)
good2 \( 1 + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 9 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.42382989853517912004077694458, −14.11125275639621900792162393084, −13.19069815363258335174666499645, −12.84688978513418512171462235404, −10.23827402727964604848387184191, −9.642727340101070443548067847258, −8.618966621189813626434174913816, −6.66463575826783700250676263362, −5.14306131647545403565477997005, −3.00632648437536992753290178800, 3.00632648437536992753290178800, 5.14306131647545403565477997005, 6.66463575826783700250676263362, 8.618966621189813626434174913816, 9.642727340101070443548067847258, 10.23827402727964604848387184191, 12.84688978513418512171462235404, 13.19069815363258335174666499645, 14.11125275639621900792162393084, 15.42382989853517912004077694458

Graph of the $Z$-function along the critical line