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Properties

Label 1-173-173.65-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: $\chi_{173} (65, \cdot )$
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

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

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