↑

Properties

Label 1-173-173.122-r0-0-0
Degree $1$
Conductor $173$
Sign $-0.251 + 0.967i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.872 + 0.489i)2-s + (0.322 + 0.946i)3-s + (0.520 + 0.853i)4-s + (0.457 + 0.889i)5-s + (−0.181 + 0.983i)6-s + (−0.109 − 0.994i)7-s + (0.0365 + 0.999i)8-s + (−0.791 + 0.611i)9-s + (−0.0365 + 0.999i)10-s + (0.694 − 0.719i)11-s + (−0.639 + 0.768i)12-s + (−0.872 − 0.489i)13-s + (0.391 − 0.920i)14-s + (−0.694 + 0.719i)15-s + (−0.457 + 0.889i)16-s + (0.934 − 0.357i)17-s + ⋯
L(s)  = 1  + (0.872 + 0.489i)2-s + (0.322 + 0.946i)3-s + (0.520 + 0.853i)4-s + (0.457 + 0.889i)5-s + (−0.181 + 0.983i)6-s + (−0.109 − 0.994i)7-s + (0.0365 + 0.999i)8-s + (−0.791 + 0.611i)9-s + (−0.0365 + 0.999i)10-s + (0.694 − 0.719i)11-s + (−0.639 + 0.768i)12-s + (−0.872 − 0.489i)13-s + (0.391 − 0.920i)14-s + (−0.694 + 0.719i)15-s + (−0.457 + 0.889i)16-s + (0.934 − 0.357i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.251 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.251 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(173\)
Sign: $-0.251 + 0.967i$
Analytic conductor: \(0.803408\)
Root analytic conductor: \(0.803408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{173} (122, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 173,\ (0:\ ),\ -0.251 + 0.967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.270357678 + 1.642570224i\)
\(L(\frac12)\) \(\approx\) \(1.270357678 + 1.642570224i\)
\(L(1)\) \(\approx\) \(1.469736217 + 1.092699214i\)
\(L(1)\) \(\approx\) \(1.469736217 + 1.092699214i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad173 \( 1 \)
good2 \( 1 + (0.872 + 0.489i)T \)
3 \( 1 + (0.322 + 0.946i)T \)
5 \( 1 + (0.457 + 0.889i)T \)
7 \( 1 + (-0.109 - 0.994i)T \)
11 \( 1 + (0.694 - 0.719i)T \)
13 \( 1 + (-0.872 - 0.489i)T \)
17 \( 1 + (0.934 - 0.357i)T \)
19 \( 1 + (-0.391 - 0.920i)T \)
23 \( 1 + (-0.694 - 0.719i)T \)
29 \( 1 + (-0.181 - 0.983i)T \)
31 \( 1 + (-0.322 + 0.946i)T \)
37 \( 1 + (0.391 + 0.920i)T \)
41 \( 1 + (0.109 + 0.994i)T \)
43 \( 1 + (0.520 - 0.853i)T \)
47 \( 1 + (0.252 + 0.967i)T \)
53 \( 1 + (-0.744 - 0.667i)T \)
59 \( 1 + (0.976 - 0.217i)T \)
61 \( 1 + (0.934 + 0.357i)T \)
67 \( 1 + (-0.322 - 0.946i)T \)
71 \( 1 + (0.181 + 0.983i)T \)
73 \( 1 + (-0.997 - 0.0729i)T \)
79 \( 1 + (-0.252 + 0.967i)T \)
83 \( 1 + (-0.581 - 0.813i)T \)
89 \( 1 + (0.833 + 0.551i)T \)
97 \( 1 + (0.581 - 0.813i)T \)
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

−27.75979848349994233389018041745, −25.606612060036201745861951816540, −25.107542776266250008121365784047, −24.29869378561483349915364479456, −23.51275302737532952791192442140, −22.31413474819314088668784986808, −21.39635681054073982571641661210, −20.42974693651396313116842483271, −19.5496227715715359119462265735, −18.75411640621645173300451173230, −17.51426063399635018982085519688, −16.32924724747663590525719853902, −14.79949450182182544464620435116, −14.27983753957917392281176316138, −12.94683162303661204793986094050, −12.32554247623093074141092231172, −11.79249636382518961235220183104, −9.84090996558192940489970453450, −9.02670407546333317141038795940, −7.52626797992637351842997336734, −6.14645850944774607552615741020, −5.39680843224976774051449140545, −3.894581676702334905584863739850, −2.2715974612265686343663244985, −1.57023176168260540968036169590, 2.64820498955518743903651811688, 3.52612509529381847674827249534, 4.62342182125707378796305937400, 5.8877068850920318192907238380, 7.00173242731767653325949247875, 8.13201014448101619874324363796, 9.68977012922292364670832157610, 10.65119809359016367712241180491, 11.614982310185025617672625721741, 13.24210854964416502020944856577, 14.26233886258258733846867901090, 14.56726057105511091298606136859, 15.8073014200633182204856886714, 16.81233818743529515246511941944, 17.48071356711974370216385344645, 19.311366146362931594314724699664, 20.2793703822271627054310501251, 21.27817245309344948131337555843, 22.155884980504480464727021313980, 22.63133417093406488051354345296, 23.80411124202315762541348909248, 25.046469784004704118278104016906, 25.82939510488310866830564713477, 26.697333031665649443640748234922, 27.24035648476912753742142037421

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