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Properties

Label 1-173-173.25-r0-0-0
Degree $1$
Conductor $173$
Sign $0.203 + 0.979i$
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  + (−0.957 + 0.288i)2-s + (0.0365 + 0.999i)3-s + (0.833 − 0.551i)4-s + (−0.391 − 0.920i)5-s + (−0.322 − 0.946i)6-s + (0.872 + 0.489i)7-s + (−0.639 + 0.768i)8-s + (−0.997 + 0.0729i)9-s + (0.639 + 0.768i)10-s + (−0.905 + 0.424i)11-s + (0.581 + 0.813i)12-s + (0.957 − 0.288i)13-s + (−0.976 − 0.217i)14-s + (0.905 − 0.424i)15-s + (0.391 − 0.920i)16-s + (0.791 + 0.611i)17-s + ⋯
L(s)  = 1  + (−0.957 + 0.288i)2-s + (0.0365 + 0.999i)3-s + (0.833 − 0.551i)4-s + (−0.391 − 0.920i)5-s + (−0.322 − 0.946i)6-s + (0.872 + 0.489i)7-s + (−0.639 + 0.768i)8-s + (−0.997 + 0.0729i)9-s + (0.639 + 0.768i)10-s + (−0.905 + 0.424i)11-s + (0.581 + 0.813i)12-s + (0.957 − 0.288i)13-s + (−0.976 − 0.217i)14-s + (0.905 − 0.424i)15-s + (0.391 − 0.920i)16-s + (0.791 + 0.611i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.203 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.203 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(173\)
Sign: $0.203 + 0.979i$
Analytic conductor: \(0.803408\)
Root analytic conductor: \(0.803408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{173} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 173,\ (0:\ ),\ 0.203 + 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5722258709 + 0.4656640561i\)
\(L(\frac12)\) \(\approx\) \(0.5722258709 + 0.4656640561i\)
\(L(1)\) \(\approx\) \(0.6719150131 + 0.2882059633i\)
\(L(1)\) \(\approx\) \(0.6719150131 + 0.2882059633i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad173 \( 1 \)
good2 \( 1 + (-0.957 + 0.288i)T \)
3 \( 1 + (0.0365 + 0.999i)T \)
5 \( 1 + (-0.391 - 0.920i)T \)
7 \( 1 + (0.872 + 0.489i)T \)
11 \( 1 + (-0.905 + 0.424i)T \)
13 \( 1 + (0.957 - 0.288i)T \)
17 \( 1 + (0.791 + 0.611i)T \)
19 \( 1 + (0.976 - 0.217i)T \)
23 \( 1 + (0.905 + 0.424i)T \)
29 \( 1 + (-0.322 + 0.946i)T \)
31 \( 1 + (-0.0365 + 0.999i)T \)
37 \( 1 + (-0.976 + 0.217i)T \)
41 \( 1 + (-0.872 - 0.489i)T \)
43 \( 1 + (0.833 + 0.551i)T \)
47 \( 1 + (0.989 + 0.145i)T \)
53 \( 1 + (-0.252 + 0.967i)T \)
59 \( 1 + (-0.520 - 0.853i)T \)
61 \( 1 + (0.791 - 0.611i)T \)
67 \( 1 + (-0.0365 - 0.999i)T \)
71 \( 1 + (0.322 - 0.946i)T \)
73 \( 1 + (-0.181 + 0.983i)T \)
79 \( 1 + (-0.989 + 0.145i)T \)
83 \( 1 + (-0.694 - 0.719i)T \)
89 \( 1 + (0.109 + 0.994i)T \)
97 \( 1 + (0.694 - 0.719i)T \)
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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

−27.12887042057825332192144606605, −26.40505074405287855806265754558, −25.57774650600494197693420332896, −24.46970665955675868280805492686, −23.610203509849076804877279862491, −22.64192467718762489365121693597, −20.99636374682601596982360587749, −20.43317963214816714490957361964, −19.00061361638908551991970662917, −18.61557765097256620700623846086, −17.86930214512050168830372198288, −16.77287361059330556159100445931, −15.56652540358994097668962546642, −14.27900310766520548300451710140, −13.30417593313074629277986360271, −11.76944089027065031900512530491, −11.25757725185674997008379715969, −10.26824240496606421612359721470, −8.61648574299304417610566040778, −7.69588190233467609980406475056, −7.1440972537617131171951301170, −5.77767408987720132746855224747, −3.4946172399278731816910464307, −2.3850494581519363291506924771, −0.94071450651963791591520181573, 1.42061575671651626262732829237, 3.23992309625744862957357312136, 5.02050916177916736819581068, 5.56014257982260803397262233322, 7.607663626267625604401417685419, 8.48953802199588338057695438046, 9.18460045823212589860490705306, 10.45363454629760881977113249825, 11.26971243084890735720721574762, 12.39965456474128176861450063061, 14.17548916717930915874011742206, 15.47964968077207091420256949849, 15.71962051714882993611206901203, 16.88351488099942868004949463685, 17.7358740941271461090938377732, 18.828748952975177273858254973714, 20.17782660621375104801027508414, 20.72232538778436005848126662511, 21.440372387461593438846210661268, 23.19142180738564918025648476520, 23.931135574357974618071868479061, 25.11526715810165447999835698369, 25.841850952876023109751765715639, 26.9618257654920119572020285623, 27.724242963582585245454806788512

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