Properties

Degree 1
Conductor 173
Sign $-0.243 + 0.969i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.581 + 0.813i)2-s + (0.872 + 0.489i)3-s + (−0.322 + 0.946i)4-s + (0.791 − 0.611i)5-s + (0.109 + 0.994i)6-s + (−0.639 + 0.768i)7-s + (−0.957 + 0.288i)8-s + (0.520 + 0.853i)9-s + (0.957 + 0.288i)10-s + (−0.989 + 0.145i)11-s + (−0.744 + 0.667i)12-s + (−0.581 − 0.813i)13-s + (−0.997 − 0.0729i)14-s + (0.989 − 0.145i)15-s + (−0.791 − 0.611i)16-s + (0.976 + 0.217i)17-s + ⋯
L(s,χ)  = 1  + (0.581 + 0.813i)2-s + (0.872 + 0.489i)3-s + (−0.322 + 0.946i)4-s + (0.791 − 0.611i)5-s + (0.109 + 0.994i)6-s + (−0.639 + 0.768i)7-s + (−0.957 + 0.288i)8-s + (0.520 + 0.853i)9-s + (0.957 + 0.288i)10-s + (−0.989 + 0.145i)11-s + (−0.744 + 0.667i)12-s + (−0.581 − 0.813i)13-s + (−0.997 − 0.0729i)14-s + (0.989 − 0.145i)15-s + (−0.791 − 0.611i)16-s + (0.976 + 0.217i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.243 + 0.969i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.243 + 0.969i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(173\)
\( \varepsilon \)  =  $-0.243 + 0.969i$
motivic weight  =  \(0\)
character  :  $\chi_{173} (88, \cdot )$
Sato-Tate  :  $\mu(86)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 173,\ (0:\ ),\ -0.243 + 0.969i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(1.185220564 + 1.519301544i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(1.185220564 + 1.519301544i\)
\(L(\chi,1)\)  \(\approx\)  \(1.376859025 + 1.018853903i\)
\(L(1,\chi)\)  \(\approx\)  \(1.376859025 + 1.018853903i\)

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

−26.95620987372853710272902244641, −26.303408937767984644944337764597, −25.33149523909224000562560193536, −24.157589464507731289419826160002, −23.31554558028226371156977437151, −22.29522436555607161775844790140, −21.17477800305630919610339818990, −20.587074767162711896486760403502, −19.41226693764532170608555683617, −18.75876007082199190792491228585, −17.87663976238228239571598640717, −16.24159992697256837081112031626, −14.72860775175243193899470679053, −14.11321147922737788375769454741, −13.29630807008117585018520833392, −12.54108216352580553097398109816, −11.03315358534648424169188230596, −9.909571688847935509756954360206, −9.352834597053139331974958036636, −7.49120664095996775686344731162, −6.52200501468711875131742381943, −5.10791669840614494115391703804, −3.42119929440733875976379204416, −2.76707989624232075050315044209, −1.41890126562951118149331681088, 2.466694650808347717626170642761, 3.38913041098551290408437755579, 5.13253514175467274421960175294, 5.527834531005447734730188047748, 7.28437767917090707041579158533, 8.3415250079858183870622278833, 9.305746214282878657091347605252, 10.17144371846844452536759400600, 12.32935722403120893269699444976, 13.04183570060466062281574987586, 13.89794111815396152772318826808, 15.03564600580391063397161887974, 15.75587156544490413328559665133, 16.6286027130467218104345116381, 17.80486873329116283792153481098, 18.949641605140306289051438435808, 20.405734962011193204957675286766, 21.14273897459512682037415043883, 21.91039241202978129097750186594, 22.85393893226769948886046952337, 24.26573256098687337257393488214, 25.084093161580200076656886998853, 25.561125301541641742008768095240, 26.43471068356215376169268357504, 27.53334256547193206272212060259

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