Properties

Degree 1
Conductor 53
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s + 11-s − 12-s + 13-s − 14-s + 15-s + 16-s + 17-s − 18-s − 19-s − 20-s − 21-s − 22-s − 23-s + 24-s + 25-s − 26-s − 27-s + 28-s + ⋯
L(s,χ)  = 1  − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s + 11-s − 12-s + 13-s − 14-s + 15-s + 16-s + 17-s − 18-s − 19-s − 20-s − 21-s − 22-s − 23-s + 24-s + 25-s − 26-s − 27-s + 28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(53\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{53} (52, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(1,\ 53,\ (0:\ ),\ 1)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.4413243698$
$L(\frac12,\chi)$  $\approx$  $0.4413243698$
$L(\chi,1)$  $\approx$  0.5400249451
$L(1,\chi)$  $\approx$  0.5400249451

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−33.86152436170425418701982870885, −32.57269619264754611232473349504, −30.545620946045299761643330146871, −29.966602467693917263919139438815, −28.36648787304449427243069888533, −27.602954185546719470641618338062, −27.13416327234771392576660466512, −25.42092095490063799611210147225, −24.05560065587156226990139008675, −23.35307579450127140854381574674, −21.66216708591945991738451762447, −20.4164355781754680603999999097, −19.05859966143852274240922060246, −18.0793699649316956247127383474, −16.94426102004003420824110155822, −16.00289465879198906094236017551, −14.73731489654954081513575792093, −12.20791887663061997561285230493, −11.45107574999535622633355811796, −10.48461614494641777812922491700, −8.639675813364645526117778754596, −7.44948828207489986312977424102, −6.04706264523221344011253005798, −4.0857712719475633052890036949, −1.28711359688044475280006335894, 1.28711359688044475280006335894, 4.0857712719475633052890036949, 6.04706264523221344011253005798, 7.44948828207489986312977424102, 8.639675813364645526117778754596, 10.48461614494641777812922491700, 11.45107574999535622633355811796, 12.20791887663061997561285230493, 14.73731489654954081513575792093, 16.00289465879198906094236017551, 16.94426102004003420824110155822, 18.0793699649316956247127383474, 19.05859966143852274240922060246, 20.4164355781754680603999999097, 21.66216708591945991738451762447, 23.35307579450127140854381574674, 24.05560065587156226990139008675, 25.42092095490063799611210147225, 27.13416327234771392576660466512, 27.602954185546719470641618338062, 28.36648787304449427243069888533, 29.966602467693917263919139438815, 30.545620946045299761643330146871, 32.57269619264754611232473349504, 33.86152436170425418701982870885

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