Properties

Degree 1
Conductor 13
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 + 14-s − 15-s + 16-s + 17-s − 18-s − 19-s − 20-s − 21-s + 22-s + 23-s − 24-s + 25-s + 27-s − 28-s + 29-s + 30-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 + 14-s − 15-s + 16-s + 17-s − 18-s − 19-s − 20-s − 21-s + 22-s + 23-s − 24-s + 25-s + 27-s − 28-s + 29-s + 30-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 13 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(13\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{13} (12, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(1,\ 13,\ (0:\ ),\ 1)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.4395929735$
$L(\frac12,\chi)$  $\approx$  $0.4395929735$
$L(\chi,1)$  $\approx$  0.6627353910
$L(1,\chi)$  $\approx$  0.6627353910

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

−43.988308502403416504944841263, −42.788261511645281673975244898140, −41.95272842208549146545251634782, −39.03678471619023999728500556745, −38.38532547892825026671651055760, −36.727547051537073200591290509603, −35.76319303421611596021350693284, −34.46644233774365394408596282416, −32.4234172849637336609819268055, −31.01874209224480391870730181195, −29.39145603391187839555762592893, −27.64980842472696812713008959723, −26.38431352575075899278988900685, −25.37170440577142621541053358349, −23.59202788517293715038687724139, −20.95918191740503118143937053273, −19.54804144334703986992466920641, −18.75125235623423291645518570775, −16.2748260574985884178954092949, −15.14833241700574422460628551477, −12.61701279102317873036853708840, −10.33642072623153902983172482625, −8.62542663503259159734490395156, −7.23159073941876201502754170862, −3.119341479008603413901599756715, 3.119341479008603413901599756715, 7.23159073941876201502754170862, 8.62542663503259159734490395156, 10.33642072623153902983172482625, 12.61701279102317873036853708840, 15.14833241700574422460628551477, 16.2748260574985884178954092949, 18.75125235623423291645518570775, 19.54804144334703986992466920641, 20.95918191740503118143937053273, 23.59202788517293715038687724139, 25.37170440577142621541053358349, 26.38431352575075899278988900685, 27.64980842472696812713008959723, 29.39145603391187839555762592893, 31.01874209224480391870730181195, 32.4234172849637336609819268055, 34.46644233774365394408596282416, 35.76319303421611596021350693284, 36.727547051537073200591290509603, 38.38532547892825026671651055760, 39.03678471619023999728500556745, 41.95272842208549146545251634782, 42.788261511645281673975244898140, 43.988308502403416504944841263

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