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Properties

Label 1-173-173.22-r0-0-0
Degree $1$
Conductor $173$
Sign $-0.344 + 0.938i$
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  + (0.639 + 0.768i)2-s + (0.109 − 0.994i)3-s + (−0.181 + 0.983i)4-s + (−0.934 + 0.357i)5-s + (0.833 − 0.551i)6-s + (−0.0365 + 0.999i)7-s + (−0.872 + 0.489i)8-s + (−0.976 − 0.217i)9-s + (−0.872 − 0.489i)10-s + (0.252 + 0.967i)11-s + (0.957 + 0.288i)12-s + (0.639 + 0.768i)13-s + (−0.791 + 0.611i)14-s + (0.252 + 0.967i)15-s + (−0.934 − 0.357i)16-s + (0.391 + 0.920i)17-s + ⋯
L(s)  = 1  + (0.639 + 0.768i)2-s + (0.109 − 0.994i)3-s + (−0.181 + 0.983i)4-s + (−0.934 + 0.357i)5-s + (0.833 − 0.551i)6-s + (−0.0365 + 0.999i)7-s + (−0.872 + 0.489i)8-s + (−0.976 − 0.217i)9-s + (−0.872 − 0.489i)10-s + (0.252 + 0.967i)11-s + (0.957 + 0.288i)12-s + (0.639 + 0.768i)13-s + (−0.791 + 0.611i)14-s + (0.252 + 0.967i)15-s + (−0.934 − 0.357i)16-s + (0.391 + 0.920i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.344 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.344 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(173\)
Sign: $-0.344 + 0.938i$
Analytic conductor: \(0.803408\)
Root analytic conductor: \(0.803408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{173} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 173,\ (0:\ ),\ -0.344 + 0.938i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6745744681 + 0.9656623473i\)
\(L(\frac12)\) \(\approx\) \(0.6745744681 + 0.9656623473i\)
\(L(1)\) \(\approx\) \(1.015170790 + 0.5708551022i\)
\(L(1)\) \(\approx\) \(1.015170790 + 0.5708551022i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad173 \( 1 \)
good2 \( 1 + (0.639 + 0.768i)T \)
3 \( 1 + (0.109 - 0.994i)T \)
5 \( 1 + (-0.934 + 0.357i)T \)
7 \( 1 + (-0.0365 + 0.999i)T \)
11 \( 1 + (0.252 + 0.967i)T \)
13 \( 1 + (0.639 + 0.768i)T \)
17 \( 1 + (0.391 + 0.920i)T \)
19 \( 1 + (-0.791 - 0.611i)T \)
23 \( 1 + (0.252 - 0.967i)T \)
29 \( 1 + (0.833 + 0.551i)T \)
31 \( 1 + (0.109 + 0.994i)T \)
37 \( 1 + (-0.791 - 0.611i)T \)
41 \( 1 + (-0.0365 + 0.999i)T \)
43 \( 1 + (-0.181 - 0.983i)T \)
47 \( 1 + (0.905 - 0.424i)T \)
53 \( 1 + (-0.694 - 0.719i)T \)
59 \( 1 + (-0.997 - 0.0729i)T \)
61 \( 1 + (0.391 - 0.920i)T \)
67 \( 1 + (0.109 - 0.994i)T \)
71 \( 1 + (0.833 + 0.551i)T \)
73 \( 1 + (0.520 + 0.853i)T \)
79 \( 1 + (0.905 + 0.424i)T \)
83 \( 1 + (0.744 + 0.667i)T \)
89 \( 1 + (-0.322 + 0.946i)T \)
97 \( 1 + (0.744 - 0.667i)T \)
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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

−27.41326944932709549141797859465, −26.71557355829841199933411508502, −25.22635402254118200811381822379, −23.88875032804501907237097982892, −23.088582307876951514469622123, −22.476849248576469333079930155903, −21.11108098973841181114096615194, −20.60287252505666747048525145021, −19.71768916350258374954028841058, −18.91684749195313728367741255999, −17.16257348064347878237388544453, −16.08402859946089693677269974257, −15.33518338555927193402811281908, −14.151518434340040132670664920440, −13.3495639802429355290785429478, −11.910705825651831725117865435587, −11.06245760982908515615016448725, −10.31341319207915268483197104457, −9.07392168343024937850222534897, −7.8922338259104483859580910706, −6.02079682620947343919969023956, −4.79039372426876401818481002387, −3.81605400568627718590693562276, −3.17010753466171671523793084396, −0.79845409654905768330407718183, 2.19233407211058795470712615636, 3.49947409570890609682194731508, 4.85743365503357850495857152463, 6.404160169702775323572758391364, 6.90230243926819907428188987067, 8.23215859101632158263646314096, 8.82655867831259910308820949958, 11.11533303487180495464396944751, 12.37778612482634527162619699386, 12.48286528568688579028778314362, 14.08785188912630676116706544956, 14.87954931152094086153495654684, 15.68330583802584767578007160728, 16.94624658518643894456141300489, 18.05073550316509233626024350823, 18.857453530816915765529988312986, 19.86387339403193496362701680390, 21.23545299580568098860010661580, 22.333421918363231462606314284957, 23.32174490385800492632707502324, 23.74694987317019519911800924503, 24.88294835741333296032138402783, 25.61924097310955855015253564482, 26.394755975217036870357678219016, 27.829119446332236697915937481353

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