Properties

Label 2-57-1.1-c1-0-1
Degree $2$
Conductor $57$
Sign $1$
Analytic cond. $0.455147$
Root an. cond. $0.674646$
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  + 2-s + 3-s − 4-s − 2·5-s + 6-s − 3·8-s + 9-s − 2·10-s − 12-s + 6·13-s − 2·15-s − 16-s − 6·17-s + 18-s − 19-s + 2·20-s + 4·23-s − 3·24-s − 25-s + 6·26-s + 27-s + 2·29-s − 2·30-s + 8·31-s + 5·32-s − 6·34-s − 36-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.894·5-s + 0.408·6-s − 1.06·8-s + 1/3·9-s − 0.632·10-s − 0.288·12-s + 1.66·13-s − 0.516·15-s − 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.229·19-s + 0.447·20-s + 0.834·23-s − 0.612·24-s − 1/5·25-s + 1.17·26-s + 0.192·27-s + 0.371·29-s − 0.365·30-s + 1.43·31-s + 0.883·32-s − 1.02·34-s − 1/6·36-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: $1$
Analytic conductor: \(0.455147\)
Root analytic conductor: \(0.674646\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{57} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 57,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.085302272\)
\(L(\frac12)\) \(\approx\) \(1.085302272\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−15.35090650815041953121172131081, −13.89424674132429450373973021658, −13.30135307533368428997273937212, −12.07987440084703696834915246162, −10.84155613367776600825623791063, −9.032116632714081125535639739797, −8.263513432862467471567669978548, −6.46787589567348650226556085189, −4.56991321080597356703354311172, −3.44365518362789223021999993571, 3.44365518362789223021999993571, 4.56991321080597356703354311172, 6.46787589567348650226556085189, 8.263513432862467471567669978548, 9.032116632714081125535639739797, 10.84155613367776600825623791063, 12.07987440084703696834915246162, 13.30135307533368428997273937212, 13.89424674132429450373973021658, 15.35090650815041953121172131081

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