Properties

Degree 1
Conductor $ 2^{2} \cdot 5 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + 3-s + 7-s + 9-s − 11-s − 13-s − 17-s − 19-s + 21-s + 23-s + 27-s + 29-s − 31-s − 33-s − 37-s − 39-s + 41-s + 43-s + 47-s + 49-s − 51-s − 53-s − 57-s − 59-s + 61-s + 63-s + 67-s + 69-s + ⋯
L(s,χ)  = 1  + 3-s + 7-s + 9-s − 11-s − 13-s − 17-s − 19-s + 21-s + 23-s + 27-s + 29-s − 31-s − 33-s − 37-s − 39-s + 41-s + 43-s + 47-s + 49-s − 51-s − 53-s − 57-s − 59-s + 61-s + 63-s + 67-s + 69-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 20 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(20\)    =    \(2^{2} \cdot 5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{20} (19, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(1,\ 20,\ (1:\ ),\ 1)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.679671111$
$L(\frac12,\chi)$  $\approx$  $1.679671111$
$L(\chi,1)$  $\approx$  1.404962946
$L(1,\chi)$  $\approx$  1.404962946

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

−39.80481063438160399687206257088, −38.36118664682136497386941158032, −37.05093651805288071763400427155, −36.351014501254046329911488825377, −34.51853083486020412867075384299, −33.16200675697415676234365610618, −31.629850769137191816888066837853, −30.813432186697565045945988246015, −29.31563646294754198450981764783, −27.42941077800643637202735468112, −26.378159592740234285714437101144, −24.90661039953541278981646848032, −23.786167100158956031134321134801, −21.62220640648137065696932558496, −20.54210870445473529316857194361, −19.12025280279851150342337755208, −17.60073479127063360348826513537, −15.49353496124302680886517579200, −14.33616900750051547377242364324, −12.80240081086662892203384420634, −10.66330073495362225432062668787, −8.80452742454490020235868190318, −7.42910977458417845335920059741, −4.67550774984207957491488368557, −2.35893499408665604861501236977, 2.35893499408665604861501236977, 4.67550774984207957491488368557, 7.42910977458417845335920059741, 8.80452742454490020235868190318, 10.66330073495362225432062668787, 12.80240081086662892203384420634, 14.33616900750051547377242364324, 15.49353496124302680886517579200, 17.60073479127063360348826513537, 19.12025280279851150342337755208, 20.54210870445473529316857194361, 21.62220640648137065696932558496, 23.786167100158956031134321134801, 24.90661039953541278981646848032, 26.378159592740234285714437101144, 27.42941077800643637202735468112, 29.31563646294754198450981764783, 30.813432186697565045945988246015, 31.629850769137191816888066837853, 33.16200675697415676234365610618, 34.51853083486020412867075384299, 36.351014501254046329911488825377, 37.05093651805288071763400427155, 38.36118664682136497386941158032, 39.80481063438160399687206257088

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