Properties

Degree 1
Conductor 107
Sign $0.309 + 0.950i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.0887 − 0.996i)2-s + (−0.998 − 0.0592i)3-s + (−0.984 + 0.176i)4-s + (−0.861 − 0.508i)5-s + (0.0296 + 0.999i)6-s + (−0.630 − 0.776i)7-s + (0.263 + 0.964i)8-s + (0.992 + 0.118i)9-s + (−0.430 + 0.902i)10-s + (0.375 + 0.926i)11-s + (0.992 − 0.118i)12-s + (−0.320 + 0.947i)13-s + (−0.717 + 0.696i)14-s + (0.829 + 0.558i)15-s + (0.937 − 0.348i)16-s + (−0.533 + 0.845i)17-s + ⋯
L(s,χ)  = 1  + (−0.0887 − 0.996i)2-s + (−0.998 − 0.0592i)3-s + (−0.984 + 0.176i)4-s + (−0.861 − 0.508i)5-s + (0.0296 + 0.999i)6-s + (−0.630 − 0.776i)7-s + (0.263 + 0.964i)8-s + (0.992 + 0.118i)9-s + (−0.430 + 0.902i)10-s + (0.375 + 0.926i)11-s + (0.992 − 0.118i)12-s + (−0.320 + 0.947i)13-s + (−0.717 + 0.696i)14-s + (0.829 + 0.558i)15-s + (0.937 − 0.348i)16-s + (−0.533 + 0.845i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.309 + 0.950i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.309 + 0.950i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(107\)
\( \varepsilon \)  =  $0.309 + 0.950i$
motivic weight  =  \(0\)
character  :  $\chi_{107} (19, \cdot )$
Sato-Tate  :  $\mu(53)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 107,\ (0:\ ),\ 0.309 + 0.950i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.09740126861 + 0.07072154814i$
$L(\frac12,\chi)$  $\approx$  $0.09740126861 + 0.07072154814i$
$L(\chi,1)$  $\approx$  0.3814998063 - 0.1741573782i
$L(1,\chi)$  $\approx$  0.3814998063 - 0.1741573782i

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

−29.32110343778069053758039708330, −27.996373530530803179992379463847, −27.38761446832006962819221839491, −26.487751638427954331311345913659, −25.180421417718482110358873451402, −24.18643719450765626229516510159, −23.28137525493996154133436112525, −22.29641449991430192940776371070, −21.899987447282967610242183187642, −19.58779700818465863684419474531, −18.656694212210152546722042733944, −17.80510626499296502712295314759, −16.548213802262907760470252408646, −15.71942284138002326964809526907, −15.044165842025506076379407638461, −13.382350515413276497762432124233, −12.20104991944525724644795714178, −11.049793236363404141670025224, −9.697271027641536611022615318138, −8.32406715514808488334802845668, −6.980451787857663203548413695, −6.11231262118170671473109012205, −4.939086631335514468181336867373, −3.42762917720143136151474310936, −0.14124447179007389408222445552, 1.66903578611051739160671093480, 4.05876050166273081519834285830, 4.47645971252059642724459955380, 6.446020041056304037360280421313, 7.8519407381550100203378842014, 9.465367286335761711490726785836, 10.47033276877518683726601008529, 11.59479053325840902677283729132, 12.41688825443972831305214777733, 13.23266103702365999731074410902, 14.971465862975933916096113581724, 16.6436129114749827649715014349, 17.04937165370677307683838408391, 18.513350890188731485880232957293, 19.498741904940835991471163029908, 20.316460364388026188246863452712, 21.58644034326382280214025653308, 22.706050021599822065602994555545, 23.29262593436013593091724448123, 24.24015621818589841526630554327, 26.19924992585404681840906915240, 27.07444375652301525034471883080, 28.18002673064789473227415002291, 28.57163175870183091477102451827, 29.77111186022001784396135994153

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