Properties

Degree 1
Conductor 107
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

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(107\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{107} (106, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(1,\ 107,\ (1:\ ),\ 1)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.326852223$
$L(\frac12,\chi)$  $\approx$  $1.326852223$
$L(\chi,1)$  $\approx$  0.9111276755
$L(1,\chi)$  $\approx$  0.9111276755

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

−29.2507140111264096034499664478, −28.14960124760015303204373695397, −27.04871897087953847297680943008, −26.49471452647039048371369461478, −25.39703258965952156230327048379, −24.680981528594938389175505521, −23.39650356421593954043645065110, −21.96844673492842249095618647735, −20.44994622216287872583535436397, −19.81318600232194171014385393698, −19.12980309394917793974016224391, −18.148053187499111510642509674952, −16.44276709701150967191770631156, −15.77499870531388299215492066204, −14.82590541909745539391871807166, −13.26211254951704896426933380367, −11.97180264986610149206081297316, −10.75367821674543183729850651015, −9.304695466462708620345463212961, −8.72812555651843937650658447358, −7.41713536583272743899328295646, −6.52750358073905121597717020075, −3.9027623586238545483928887445, −2.90812216403723678026329739840, −1.01532045680624064311140232848, 1.01532045680624064311140232848, 2.90812216403723678026329739840, 3.9027623586238545483928887445, 6.52750358073905121597717020075, 7.41713536583272743899328295646, 8.72812555651843937650658447358, 9.304695466462708620345463212961, 10.75367821674543183729850651015, 11.97180264986610149206081297316, 13.26211254951704896426933380367, 14.82590541909745539391871807166, 15.77499870531388299215492066204, 16.44276709701150967191770631156, 18.148053187499111510642509674952, 19.12980309394917793974016224391, 19.81318600232194171014385393698, 20.44994622216287872583535436397, 21.96844673492842249095618647735, 23.39650356421593954043645065110, 24.680981528594938389175505521, 25.39703258965952156230327048379, 26.49471452647039048371369461478, 27.04871897087953847297680943008, 28.14960124760015303204373695397, 29.2507140111264096034499664478

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