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Properties

Label 1-173-173.8-r1-0-0
Degree $1$
Conductor $173$
Sign $-0.910 + 0.413i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.994 + 0.109i)2-s + (−0.983 + 0.181i)3-s + (0.976 + 0.217i)4-s + (−0.424 − 0.905i)5-s + (−0.997 + 0.0729i)6-s + (−0.551 − 0.833i)7-s + (0.946 + 0.322i)8-s + (0.934 − 0.357i)9-s + (−0.322 − 0.946i)10-s + (−0.813 + 0.581i)11-s + (−0.999 − 0.0365i)12-s + (−0.109 + 0.994i)13-s + (−0.457 − 0.889i)14-s + (0.581 + 0.813i)15-s + (0.905 + 0.424i)16-s + (−0.145 + 0.989i)17-s + ⋯
L(s)  = 1  + (0.994 + 0.109i)2-s + (−0.983 + 0.181i)3-s + (0.976 + 0.217i)4-s + (−0.424 − 0.905i)5-s + (−0.997 + 0.0729i)6-s + (−0.551 − 0.833i)7-s + (0.946 + 0.322i)8-s + (0.934 − 0.357i)9-s + (−0.322 − 0.946i)10-s + (−0.813 + 0.581i)11-s + (−0.999 − 0.0365i)12-s + (−0.109 + 0.994i)13-s + (−0.457 − 0.889i)14-s + (0.581 + 0.813i)15-s + (0.905 + 0.424i)16-s + (−0.145 + 0.989i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.910 + 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.910 + 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(173\)
Sign: $-0.910 + 0.413i$
Analytic conductor: \(18.5914\)
Root analytic conductor: \(18.5914\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{173} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 173,\ (1:\ ),\ -0.910 + 0.413i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07339822011 + 0.3395486559i\)
\(L(\frac12)\) \(\approx\) \(0.07339822011 + 0.3395486559i\)
\(L(1)\) \(\approx\) \(0.9690514553 + 0.04822741724i\)
\(L(1)\) \(\approx\) \(0.9690514553 + 0.04822741724i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad173 \( 1 \)
good2 \( 1 + (0.994 + 0.109i)T \)
3 \( 1 + (-0.983 + 0.181i)T \)
5 \( 1 + (-0.424 - 0.905i)T \)
7 \( 1 + (-0.551 - 0.833i)T \)
11 \( 1 + (-0.813 + 0.581i)T \)
13 \( 1 + (-0.109 + 0.994i)T \)
17 \( 1 + (-0.145 + 0.989i)T \)
19 \( 1 + (-0.889 - 0.457i)T \)
23 \( 1 + (-0.581 + 0.813i)T \)
29 \( 1 + (-0.997 - 0.0729i)T \)
31 \( 1 + (0.181 - 0.983i)T \)
37 \( 1 + (0.457 - 0.889i)T \)
41 \( 1 + (-0.833 + 0.551i)T \)
43 \( 1 + (-0.976 + 0.217i)T \)
47 \( 1 + (0.744 + 0.667i)T \)
53 \( 1 + (-0.288 - 0.957i)T \)
59 \( 1 + (-0.920 - 0.391i)T \)
61 \( 1 + (0.145 + 0.989i)T \)
67 \( 1 + (0.181 + 0.983i)T \)
71 \( 1 + (-0.0729 + 0.997i)T \)
73 \( 1 + (0.791 - 0.611i)T \)
79 \( 1 + (-0.667 - 0.744i)T \)
83 \( 1 + (0.639 + 0.768i)T \)
89 \( 1 + (-0.520 - 0.853i)T \)
97 \( 1 + (0.768 + 0.639i)T \)
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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

−26.94780507854710069580783887856, −25.595103413160796284109325294352, −24.695197541920247202965760040516, −23.61357143788268481691652369127, −22.86685436638016706482761167651, −22.21100590306164236822552031222, −21.51128935370275961736183132211, −20.15115028149867569267445042070, −18.77823994991183324429352079145, −18.385585140769903985627795424777, −16.70437722576886155384696035926, −15.70357643528542123562927294689, −15.16597760623922104918305527776, −13.71728163473549315580131351301, −12.64859795874678102392678006054, −11.91948079446938986939355896787, −10.85410912156606546466874932691, −10.190907622351688582727425886944, −7.96132932437387871135403353187, −6.76105804640822145163895321613, −5.93925805699409601406328584706, −4.986997964634549294169407449167, −3.41150041525925747057301644200, −2.35761202408386182793743326937, −0.09399309671988363753513658340, 1.735962807437369158269421835157, 3.93822864399477207621721702489, 4.47368355837757692046455074554, 5.6727324385899269999250649117, 6.78205207472300610075224478750, 7.805905729596652113071437828573, 9.68530620055124447835386866485, 10.839086683688168247667649482806, 11.769046070217677046033222234379, 12.85310202104611659678491638343, 13.277617523334160115310856621426, 15.00762054322211162964324862991, 15.89759028483498920460644702746, 16.70531001455200089576568741095, 17.33595054403532369232834281443, 19.145778433418013110154793597360, 20.16948940382184439434597886751, 21.11496065091915086543571164636, 21.939475286904308812799746772771, 23.131881966754898612565691380695, 23.68759635805750417137455018818, 24.15339182793716030597444657020, 25.70159950977646487443035814669, 26.58022919298457709262466065503, 28.12789888842360273061113967891

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