Properties

Degree $2$
Conductor $57$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3-s + 2·4-s + 5-s − 2·6-s + 3·7-s + 9-s − 2·10-s − 3·11-s + 2·12-s − 6·13-s − 6·14-s + 15-s − 4·16-s + 3·17-s − 2·18-s − 19-s + 2·20-s + 3·21-s + 6·22-s + 4·23-s − 4·25-s + 12·26-s + 27-s + 6·28-s − 10·29-s − 2·30-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.577·3-s + 4-s + 0.447·5-s − 0.816·6-s + 1.13·7-s + 1/3·9-s − 0.632·10-s − 0.904·11-s + 0.577·12-s − 1.66·13-s − 1.60·14-s + 0.258·15-s − 16-s + 0.727·17-s − 0.471·18-s − 0.229·19-s + 0.447·20-s + 0.654·21-s + 1.27·22-s + 0.834·23-s − 4/5·25-s + 2.35·26-s + 0.192·27-s + 1.13·28-s − 1.85·29-s − 0.365·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(57\)    =    \(3 \cdot 19\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{57} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 57,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5865095916\)
\(L(\frac12)\) \(\approx\) \(0.5865095916\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

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

−19.11984794703026, −18.40712597719815, −17.41386108626453, −16.78947561830285, −15.26425654515921, −14.35869809327675, −13.04832504304323, −11.43002930912672, −10.20230313909289, −9.400588261603299, −8.062798749262724, −7.423199811314048, −5.029930777904474, −2.133072577244063, 2.133072577244063, 5.029930777904474, 7.423199811314048, 8.062798749262724, 9.400588261603299, 10.20230313909289, 11.43002930912672, 13.04832504304323, 14.35869809327675, 15.26425654515921, 16.78947561830285, 17.41386108626453, 18.40712597719815, 19.11984794703026

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