Properties

Degree 1
Conductor $ 3 \cdot 7 \cdot 19 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + 2-s + 4-s + 5-s + 8-s + 10-s − 11-s + 13-s + 16-s + 17-s + 20-s − 22-s − 23-s + 25-s + 26-s + 29-s + 31-s + 32-s + 34-s − 37-s + 40-s − 41-s + 43-s − 44-s − 46-s + 47-s + 50-s + 52-s + ⋯
L(s,χ)  = 1  + 2-s + 4-s + 5-s + 8-s + 10-s − 11-s + 13-s + 16-s + 17-s + 20-s − 22-s − 23-s + 25-s + 26-s + 29-s + 31-s + 32-s + 34-s − 37-s + 40-s − 41-s + 43-s − 44-s − 46-s + 47-s + 50-s + 52-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 399 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(399\)    =    \(3 \cdot 7 \cdot 19\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{399} (398, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(1,\ 399,\ (1:\ ),\ 1)$
$L(\chi,\frac{1}{2})$  $\approx$  $5.208816059$
$L(\frac12,\chi)$  $\approx$  $5.208816059$
$L(\chi,1)$  $\approx$  2.516421618
$L(1,\chi)$  $\approx$  2.516421618

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

−24.02983373798208723313047064245, −23.275179088878022882724992758905, −22.476820200368340016599867681462, −21.391049656634069759785268753509, −21.045560088799868175579644058865, −20.22022303082149547194313819092, −18.94007971310062555682797551709, −18.056599714025826048033692494080, −17.00189402642535769248290944899, −16.045510048137678013336228436273, −15.335692809812842846592976802851, −13.99987573933064718146395033924, −13.75102620653019787121842709434, −12.71249545746190727107487456238, −11.88039341137774566413691658073, −10.546082323040423555609661501584, −10.14876321055179766661047285626, −8.59293333640678508032185656008, −7.506444097872896496429608772966, −6.26338852788059394097886376727, −5.669428063996822259556611055419, −4.68444059445583810219462029407, −3.36609594969847143044930398516, −2.39990463223900493627808874225, −1.23781436250339339745554163794, 1.23781436250339339745554163794, 2.39990463223900493627808874225, 3.36609594969847143044930398516, 4.68444059445583810219462029407, 5.669428063996822259556611055419, 6.26338852788059394097886376727, 7.506444097872896496429608772966, 8.59293333640678508032185656008, 10.14876321055179766661047285626, 10.546082323040423555609661501584, 11.88039341137774566413691658073, 12.71249545746190727107487456238, 13.75102620653019787121842709434, 13.99987573933064718146395033924, 15.335692809812842846592976802851, 16.045510048137678013336228436273, 17.00189402642535769248290944899, 18.056599714025826048033692494080, 18.94007971310062555682797551709, 20.22022303082149547194313819092, 21.045560088799868175579644058865, 21.391049656634069759785268753509, 22.476820200368340016599867681462, 23.275179088878022882724992758905, 24.02983373798208723313047064245

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