↑

Properties

Label 1-173-173.28-r1-0-0
Degree $1$
Conductor $173$
Sign $-0.657 + 0.753i$
Analytic cond. $18.5914$
Root an. cond. $18.5914$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.920 − 0.391i)2-s + (0.145 + 0.989i)3-s + (0.694 + 0.719i)4-s + (0.999 − 0.0365i)5-s + (0.252 − 0.967i)6-s + (−0.889 − 0.457i)7-s + (−0.357 − 0.934i)8-s + (−0.957 + 0.288i)9-s + (−0.934 − 0.357i)10-s + (0.983 − 0.181i)11-s + (−0.611 + 0.791i)12-s + (−0.391 + 0.920i)13-s + (0.639 + 0.768i)14-s + (0.181 + 0.983i)15-s + (−0.0365 + 0.999i)16-s + (0.489 + 0.872i)17-s + ⋯
L(s)  = 1  + (−0.920 − 0.391i)2-s + (0.145 + 0.989i)3-s + (0.694 + 0.719i)4-s + (0.999 − 0.0365i)5-s + (0.252 − 0.967i)6-s + (−0.889 − 0.457i)7-s + (−0.357 − 0.934i)8-s + (−0.957 + 0.288i)9-s + (−0.934 − 0.357i)10-s + (0.983 − 0.181i)11-s + (−0.611 + 0.791i)12-s + (−0.391 + 0.920i)13-s + (0.639 + 0.768i)14-s + (0.181 + 0.983i)15-s + (−0.0365 + 0.999i)16-s + (0.489 + 0.872i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.657 + 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.657 + 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(173\)
Sign: $-0.657 + 0.753i$
Analytic conductor: \(18.5914\)
Root analytic conductor: \(18.5914\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{173} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 173,\ (1:\ ),\ -0.657 + 0.753i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3642698653 + 0.8016429574i\)
\(L(\frac12)\) \(\approx\) \(0.3642698653 + 0.8016429574i\)
\(L(1)\) \(\approx\) \(0.7021549252 + 0.2440833841i\)
\(L(1)\) \(\approx\) \(0.7021549252 + 0.2440833841i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad173 \( 1 \)
good2 \( 1 + (-0.920 - 0.391i)T \)
3 \( 1 + (0.145 + 0.989i)T \)
5 \( 1 + (0.999 - 0.0365i)T \)
7 \( 1 + (-0.889 - 0.457i)T \)
11 \( 1 + (0.983 - 0.181i)T \)
13 \( 1 + (-0.391 + 0.920i)T \)
17 \( 1 + (0.489 + 0.872i)T \)
19 \( 1 + (-0.768 - 0.639i)T \)
23 \( 1 + (-0.181 + 0.983i)T \)
29 \( 1 + (0.252 + 0.967i)T \)
31 \( 1 + (-0.989 - 0.145i)T \)
37 \( 1 + (-0.639 + 0.768i)T \)
41 \( 1 + (0.457 - 0.889i)T \)
43 \( 1 + (-0.694 + 0.719i)T \)
47 \( 1 + (0.833 + 0.551i)T \)
53 \( 1 + (-0.853 + 0.520i)T \)
59 \( 1 + (0.813 - 0.581i)T \)
61 \( 1 + (-0.489 + 0.872i)T \)
67 \( 1 + (-0.989 + 0.145i)T \)
71 \( 1 + (-0.967 + 0.252i)T \)
73 \( 1 + (-0.744 - 0.667i)T \)
79 \( 1 + (0.551 + 0.833i)T \)
83 \( 1 + (-0.997 - 0.0729i)T \)
89 \( 1 + (-0.905 + 0.424i)T \)
97 \( 1 + (0.0729 + 0.997i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−26.72881810790569094435255862084, −25.549218603694904497424227922230, −25.122028907083737129488448486156, −24.63766452135422384595312526949, −23.14837977844563569505092236744, −22.30475193224636935934609827490, −20.71152475497860897049313764224, −19.78552863652143677164600266011, −18.88263341528991006293198436847, −18.131693639362255093805143529466, −17.229362061638354200399242433, −16.44983349756701676306389032881, −14.91302282837159196729502814406, −14.13133333499864180543183962184, −12.82518740277327472677669445902, −11.93770507831579007507234229327, −10.366284402784192650199299602327, −9.4299525749025960829559286457, −8.552635738923064414501806232298, −7.215615191566446562500691340505, −6.32445364898720377726458206072, −5.59587089923528420192189734893, −2.86796526039182841664353323019, −1.84250993635424624363468746183, −0.40465684945597248549698430960, 1.59197674117259937311751008072, 3.0559499802673864303037984657, 4.14209735796677394924295537142, 5.97027033837016053181126874661, 6.99581453796574113138092165244, 8.79521709720453852537040676822, 9.38776278221775616991032333277, 10.18421523671116699448144647207, 11.08581385292009935467776218609, 12.4261941011134390743281683546, 13.71316967856565922693852456860, 14.83936467464603844062827909947, 16.2131831027603953239154008000, 16.89958404479094381265140532294, 17.48062812812312589720838193560, 19.13018378166896978167498266324, 19.7231434947555950246029197778, 20.79411020650415425438930528065, 21.81572275372624903345953042533, 22.091152517753016292095981126938, 23.86917695428259406162759830629, 25.45291177520024256745452342903, 25.73354950576171079747844141368, 26.61737697493692257734519757401, 27.62081079499291007666014340675

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