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Properties

Label 1-161-161.89-r0-0-0
Degree $1$
Conductor $161$
Sign $-0.950 - 0.310i$
Analytic cond. $0.747680$
Root an. cond. $0.747680$
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  + (0.723 + 0.690i)2-s + (−0.981 + 0.189i)3-s + (0.0475 + 0.998i)4-s + (−0.786 + 0.618i)5-s + (−0.841 − 0.540i)6-s + (−0.654 + 0.755i)8-s + (0.928 − 0.371i)9-s + (−0.995 − 0.0950i)10-s + (−0.723 + 0.690i)11-s + (−0.235 − 0.971i)12-s + (−0.415 − 0.909i)13-s + (0.654 − 0.755i)15-s + (−0.995 + 0.0950i)16-s + (−0.888 + 0.458i)17-s + (0.928 + 0.371i)18-s + (−0.888 − 0.458i)19-s + ⋯
L(s)  = 1  + (0.723 + 0.690i)2-s + (−0.981 + 0.189i)3-s + (0.0475 + 0.998i)4-s + (−0.786 + 0.618i)5-s + (−0.841 − 0.540i)6-s + (−0.654 + 0.755i)8-s + (0.928 − 0.371i)9-s + (−0.995 − 0.0950i)10-s + (−0.723 + 0.690i)11-s + (−0.235 − 0.971i)12-s + (−0.415 − 0.909i)13-s + (0.654 − 0.755i)15-s + (−0.995 + 0.0950i)16-s + (−0.888 + 0.458i)17-s + (0.928 + 0.371i)18-s + (−0.888 − 0.458i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.950 - 0.310i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.950 - 0.310i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $-0.950 - 0.310i$
Analytic conductor: \(0.747680\)
Root analytic conductor: \(0.747680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (0:\ ),\ -0.950 - 0.310i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.09158741182 + 0.5760324805i\)
\(L(\frac12)\) \(\approx\) \(-0.09158741182 + 0.5760324805i\)
\(L(1)\) \(\approx\) \(0.5524928364 + 0.5468138397i\)
\(L(1)\) \(\approx\) \(0.5524928364 + 0.5468138397i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.723 + 0.690i)T \)
3 \( 1 + (-0.981 + 0.189i)T \)
5 \( 1 + (-0.786 + 0.618i)T \)
11 \( 1 + (-0.723 + 0.690i)T \)
13 \( 1 + (-0.415 - 0.909i)T \)
17 \( 1 + (-0.888 + 0.458i)T \)
19 \( 1 + (-0.888 - 0.458i)T \)
29 \( 1 + (0.841 + 0.540i)T \)
31 \( 1 + (0.327 + 0.945i)T \)
37 \( 1 + (-0.928 + 0.371i)T \)
41 \( 1 + (0.142 + 0.989i)T \)
43 \( 1 + (0.654 + 0.755i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (-0.580 - 0.814i)T \)
59 \( 1 + (0.995 + 0.0950i)T \)
61 \( 1 + (0.981 + 0.189i)T \)
67 \( 1 + (-0.235 + 0.971i)T \)
71 \( 1 + (-0.959 + 0.281i)T \)
73 \( 1 + (-0.0475 - 0.998i)T \)
79 \( 1 + (-0.580 + 0.814i)T \)
83 \( 1 + (-0.142 + 0.989i)T \)
89 \( 1 + (-0.327 + 0.945i)T \)
97 \( 1 + (-0.142 - 0.989i)T \)
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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

−27.554645427061890781169330883870, −26.722999814503053951097199160623, −24.69127793440042375548897980901, −24.00615647270346098381849237339, −23.389591943653260940274521624702, −22.44425519897714927190277729004, −21.44690449250497030039279914048, −20.62724158690391206203694010345, −19.24953852759271751614569165609, −18.78540615519309731148619050349, −17.29814033245954961463378004912, −16.121514738765242655684013865969, −15.43187721262082260629576900398, −13.84580869535347980924756236154, −12.868923691332292104081510756957, −11.990160961217103874122117543004, −11.26836460188815401375453376597, −10.287891554303816450567051829761, −8.798659091809034032211333034056, −7.18091392868192988361849939826, −5.93934863165216959077805410629, −4.82127973655363672977985127016, −4.01238436744933881708008046588, −2.14681258631110245504056481378, −0.416358963728450888305555796185, 2.78015544157769115488355858311, 4.244038673123645770197788994302, 5.063447722143468346298130240948, 6.42945134274290257397878917252, 7.20532542226637555179083207118, 8.35506300699179520806393463235, 10.29387303464592315667033101027, 11.184127520910621185921527453368, 12.351021365606444857597353641362, 13.00136676674428318007190609766, 14.67792486500064004158074614788, 15.46705954063621003786911363821, 16.05853312693338258303867078546, 17.51611650292215144897648869248, 17.91934102186289148979619050749, 19.49181756685233622656328790359, 20.83572776137862444977958312575, 21.95796829468009534304603886693, 22.614008420267554303171083493907, 23.43862245212784996320352327914, 24.02046532720261768085238685670, 25.33910025426128086423721730940, 26.43458832645447992866918680601, 27.18888054697354535373428460085, 28.22321192329789717076909601302

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