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Properties

Label 1-173-173.116-r0-0-0
Degree $1$
Conductor $173$
Sign $-0.243 - 0.969i$
Analytic cond. $0.803408$
Root an. cond. $0.803408$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(173\)
Sign: $-0.243 - 0.969i$
Analytic conductor: \(0.803408\)
Root analytic conductor: \(0.803408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{173} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 173,\ (0:\ ),\ -0.243 - 0.969i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.185220564 - 1.519301544i\)
\(L(\frac12)\) \(\approx\) \(1.185220564 - 1.519301544i\)
\(L(1)\) \(\approx\) \(1.376859025 - 1.018853903i\)
\(L(1)\) \(\approx\) \(1.376859025 - 1.018853903i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad173 \( 1 \)
good2 \( 1 + (0.581 - 0.813i)T \)
3 \( 1 + (0.872 - 0.489i)T \)
5 \( 1 + (0.791 + 0.611i)T \)
7 \( 1 + (-0.639 - 0.768i)T \)
11 \( 1 + (-0.989 - 0.145i)T \)
13 \( 1 + (-0.581 + 0.813i)T \)
17 \( 1 + (0.976 - 0.217i)T \)
19 \( 1 + (0.997 + 0.0729i)T \)
23 \( 1 + (0.989 - 0.145i)T \)
29 \( 1 + (0.109 + 0.994i)T \)
31 \( 1 + (-0.872 - 0.489i)T \)
37 \( 1 + (-0.997 - 0.0729i)T \)
41 \( 1 + (0.639 + 0.768i)T \)
43 \( 1 + (-0.322 + 0.946i)T \)
47 \( 1 + (-0.457 + 0.889i)T \)
53 \( 1 + (-0.905 - 0.424i)T \)
59 \( 1 + (0.181 - 0.983i)T \)
61 \( 1 + (0.976 + 0.217i)T \)
67 \( 1 + (-0.872 + 0.489i)T \)
71 \( 1 + (-0.109 - 0.994i)T \)
73 \( 1 + (0.833 - 0.551i)T \)
79 \( 1 + (0.457 + 0.889i)T \)
83 \( 1 + (0.252 - 0.967i)T \)
89 \( 1 + (-0.0365 + 0.999i)T \)
97 \( 1 + (-0.252 - 0.967i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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