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Properties

Label 1-173-173.96-r0-0-0
Degree $1$
Conductor $173$
Sign $-0.563 + 0.826i$
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  + (0.391 + 0.920i)2-s + (0.989 + 0.145i)3-s + (−0.694 + 0.719i)4-s + (−0.0365 + 0.999i)5-s + (0.252 + 0.967i)6-s + (−0.457 − 0.889i)7-s + (−0.934 − 0.357i)8-s + (0.957 + 0.288i)9-s + (−0.934 + 0.357i)10-s + (−0.181 + 0.983i)11-s + (−0.791 + 0.611i)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 + (−0.872 − 0.489i)17-s + ⋯
L(s)  = 1  + (0.391 + 0.920i)2-s + (0.989 + 0.145i)3-s + (−0.694 + 0.719i)4-s + (−0.0365 + 0.999i)5-s + (0.252 + 0.967i)6-s + (−0.457 − 0.889i)7-s + (−0.934 − 0.357i)8-s + (0.957 + 0.288i)9-s + (−0.934 + 0.357i)10-s + (−0.181 + 0.983i)11-s + (−0.791 + 0.611i)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 + (−0.872 − 0.489i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.563 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.563 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(173\)
Sign: $-0.563 + 0.826i$
Analytic conductor: \(0.803408\)
Root analytic conductor: \(0.803408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{173} (96, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 173,\ (0:\ ),\ -0.563 + 0.826i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7573218300 + 1.432431307i\)
\(L(\frac12)\) \(\approx\) \(0.7573218300 + 1.432431307i\)
\(L(1)\) \(\approx\) \(1.102232242 + 0.9811625097i\)
\(L(1)\) \(\approx\) \(1.102232242 + 0.9811625097i\)

Euler product

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

−27.3541655296776267775620881700, −26.285005830346800728476569413942, −24.98856624787395795384841677380, −24.366319410869984578277202799248, −23.35974260298874995781511555880, −21.78315098124049721981747338284, −21.45618200664123136793553141971, −20.10111409188740608267486731017, −19.78865359912916863838749571587, −18.6940621198884997120987588456, −17.75507057456726872551148144907, −15.87671476309602859009536313855, −15.29449831822277108844338381832, −13.798578348244125583736077316951, −13.143516032984377484178872307578, −12.41543466113102143207047934876, −11.167200557594531475718048722498, −9.71252902614874308201825489206, −8.89937248192620545390343901415, −8.189868731682478369143383081906, −6.08405668576821551085180669939, −4.9504381413825644070493757695, −3.54698945213224351839394449762, −2.64148427833040035097650930343, −1.180700784167414010445415349577, 2.44239099627577057880333411528, 3.72909070170553943076057678760, 4.49969616221474090266847472088, 6.525997608241814227897748305803, 7.11859069180736863365323592004, 8.11404449608921474465227776817, 9.48115175036049112290225062714, 10.31275654339505563864572086812, 12.051626836456273387636289305694, 13.572782044076034363113388084010, 13.86252938758788516616785299061, 14.98073466518779614171954980780, 15.70610054848618973567359040491, 16.78747393683969831202616404808, 18.11384274883341316160599174766, 18.88029527228044028273790053284, 20.17312127223259864689141349634, 21.08963383742154398488637135724, 22.35266113881607423375070225777, 22.95020283126578275440691471686, 24.04531473999430302641836957997, 25.15201282862437773544398240133, 25.95708624309118840112860751993, 26.56764335275671711926100212984, 27.10907574926723463503826451395

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