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Properties

Label 1-173-173.55-r0-0-0
Degree $1$
Conductor $173$
Sign $0.723 + 0.690i$
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  + (−0.639 + 0.768i)2-s + (−0.109 − 0.994i)3-s + (−0.181 − 0.983i)4-s + (0.934 + 0.357i)5-s + (0.833 + 0.551i)6-s + (0.0365 + 0.999i)7-s + (0.872 + 0.489i)8-s + (−0.976 + 0.217i)9-s + (−0.872 + 0.489i)10-s + (−0.252 + 0.967i)11-s + (−0.957 + 0.288i)12-s + (0.639 − 0.768i)13-s + (−0.791 − 0.611i)14-s + (0.252 − 0.967i)15-s + (−0.934 + 0.357i)16-s + (−0.391 + 0.920i)17-s + ⋯
L(s)  = 1  + (−0.639 + 0.768i)2-s + (−0.109 − 0.994i)3-s + (−0.181 − 0.983i)4-s + (0.934 + 0.357i)5-s + (0.833 + 0.551i)6-s + (0.0365 + 0.999i)7-s + (0.872 + 0.489i)8-s + (−0.976 + 0.217i)9-s + (−0.872 + 0.489i)10-s + (−0.252 + 0.967i)11-s + (−0.957 + 0.288i)12-s + (0.639 − 0.768i)13-s + (−0.791 − 0.611i)14-s + (0.252 − 0.967i)15-s + (−0.934 + 0.357i)16-s + (−0.391 + 0.920i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.723 + 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.723 + 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(173\)
Sign: $0.723 + 0.690i$
Analytic conductor: \(0.803408\)
Root analytic conductor: \(0.803408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{173} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 173,\ (0:\ ),\ 0.723 + 0.690i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8287851112 + 0.3320496680i\)
\(L(\frac12)\) \(\approx\) \(0.8287851112 + 0.3320496680i\)
\(L(1)\) \(\approx\) \(0.8372143775 + 0.1898888887i\)
\(L(1)\) \(\approx\) \(0.8372143775 + 0.1898888887i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad173 \( 1 \)
good2 \( 1 + (-0.639 + 0.768i)T \)
3 \( 1 + (-0.109 - 0.994i)T \)
5 \( 1 + (0.934 + 0.357i)T \)
7 \( 1 + (0.0365 + 0.999i)T \)
11 \( 1 + (-0.252 + 0.967i)T \)
13 \( 1 + (0.639 - 0.768i)T \)
17 \( 1 + (-0.391 + 0.920i)T \)
19 \( 1 + (0.791 - 0.611i)T \)
23 \( 1 + (0.252 + 0.967i)T \)
29 \( 1 + (0.833 - 0.551i)T \)
31 \( 1 + (0.109 - 0.994i)T \)
37 \( 1 + (-0.791 + 0.611i)T \)
41 \( 1 + (-0.0365 - 0.999i)T \)
43 \( 1 + (-0.181 + 0.983i)T \)
47 \( 1 + (0.905 + 0.424i)T \)
53 \( 1 + (0.694 - 0.719i)T \)
59 \( 1 + (0.997 - 0.0729i)T \)
61 \( 1 + (-0.391 - 0.920i)T \)
67 \( 1 + (0.109 + 0.994i)T \)
71 \( 1 + (-0.833 + 0.551i)T \)
73 \( 1 + (0.520 - 0.853i)T \)
79 \( 1 + (-0.905 + 0.424i)T \)
83 \( 1 + (0.744 - 0.667i)T \)
89 \( 1 + (-0.322 - 0.946i)T \)
97 \( 1 + (-0.744 - 0.667i)T \)
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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

−27.154449031636757735490829315, −26.72039462405281794138694225989, −25.88476671876856010435160799469, −24.76540924647639762136666864762, −23.23699100556575157725927557916, −22.1729040397807436896179055384, −21.27272435167549133152651655362, −20.70460230242064840773463625784, −19.91868394702990473214830876427, −18.49349553746442007476904578873, −17.56946865273252840009335012301, −16.35780584450570749648651660598, −16.335038073252931520751777603390, −14.02439226369762917348643944343, −13.6392518695020632996697104879, −12.037640744428890130327971645, −10.844062887791423004915421524941, −10.303998011405694283584575529898, −9.19874657676357606027017666743, −8.479135764265222000685734278518, −6.74995639567458556746870474052, −5.164862657252292707135949039294, −3.99962899140215997551411135072, −2.8253752275010454762329041924, −1.05594470565378744572823963519, 1.501016421847566814075111658830, 2.55657200270692367205251135648, 5.2739560626874698348415304008, 5.98418202368248689331310141059, 6.950464546219520411858502170926, 8.07536455089267809218356505490, 9.09719743172823108832512821542, 10.20220398222652808861255848343, 11.44812678468970408097388810102, 12.89961415083158878413754053708, 13.68924400884300265684590188802, 14.90540901482567855062680249476, 15.66498872172092862305026573026, 17.41607908727704106046762368822, 17.67625198668026500047984006930, 18.51522203658629170960564748758, 19.406147651791816090384116304566, 20.62251967561010919885135829198, 22.15600107501335733302780851002, 22.95581303275314438407876338554, 24.08514836798768599245518837315, 24.93607547373347043647660139407, 25.58554044503134597235066486945, 26.1500572168081745766115874608, 27.83992678131726891594163290909

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