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Properties

Label 1-161-161.143-r0-0-0
Degree $1$
Conductor $161$
Sign $0.841 - 0.539i$
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  + (−0.888 + 0.458i)2-s + (0.786 − 0.618i)3-s + (0.580 − 0.814i)4-s + (0.723 − 0.690i)5-s + (−0.415 + 0.909i)6-s + (−0.142 + 0.989i)8-s + (0.235 − 0.971i)9-s + (−0.327 + 0.945i)10-s + (0.888 + 0.458i)11-s + (−0.0475 − 0.998i)12-s + (0.654 + 0.755i)13-s + (0.142 − 0.989i)15-s + (−0.327 − 0.945i)16-s + (−0.995 + 0.0950i)17-s + (0.235 + 0.971i)18-s + (−0.995 − 0.0950i)19-s + ⋯
L(s)  = 1  + (−0.888 + 0.458i)2-s + (0.786 − 0.618i)3-s + (0.580 − 0.814i)4-s + (0.723 − 0.690i)5-s + (−0.415 + 0.909i)6-s + (−0.142 + 0.989i)8-s + (0.235 − 0.971i)9-s + (−0.327 + 0.945i)10-s + (0.888 + 0.458i)11-s + (−0.0475 − 0.998i)12-s + (0.654 + 0.755i)13-s + (0.142 − 0.989i)15-s + (−0.327 − 0.945i)16-s + (−0.995 + 0.0950i)17-s + (0.235 + 0.971i)18-s + (−0.995 − 0.0950i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.841 - 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.841 - 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $0.841 - 0.539i$
Analytic conductor: \(0.747680\)
Root analytic conductor: \(0.747680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (0:\ ),\ 0.841 - 0.539i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.064787988 - 0.3118612114i\)
\(L(\frac12)\) \(\approx\) \(1.064787988 - 0.3118612114i\)
\(L(1)\) \(\approx\) \(1.022373289 - 0.1487508323i\)
\(L(1)\) \(\approx\) \(1.022373289 - 0.1487508323i\)

Euler product

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

−27.546075729555801677487842373511, −27.04904939260618658838247758614, −25.90141036795795446377526903385, −25.47863923421809214231885509200, −24.519595074585855145011301821288, −22.48589710837118228354681325052, −21.78208804968397660523821703984, −20.93635010736073140504408307518, −19.95920018665646245856191909957, −19.13262494240060347158535544711, −18.10142076219619595111275180292, −17.134416526709874342205749176556, −16.01027442473206296622474199487, −14.966470813184919134228319119911, −13.865547067464652453026653709805, −12.77896058413213633488815958261, −10.97098520495686943317786943295, −10.59863736564442261885261548854, −9.26771295092689233141301914239, −8.70901371726072587320665584185, −7.32681030146140225942406914851, −6.07089837806459731201490100320, −3.99836094331541478464621834383, −2.94178168529727175834405465789, −1.81042723996182493922895807005, 1.36623265050794394613747189425, 2.23184803653100761157423305404, 4.357035524239440544369192922925, 6.182517548959571698817022166237, 6.83324426923619579190186530787, 8.329473370467680032636111079601, 8.99259044198862162399962769409, 9.78799528611287064080681631358, 11.35437003935056648733407298682, 12.69374405350260919316125783989, 13.80395184992698957856340173625, 14.67817725949735796257640319326, 15.83416572194718150852229251846, 17.07073927274421777033120395331, 17.7361059337883749154334931211, 18.8054417510420332976577007214, 19.75057640763647737454273454596, 20.48181951895057951601192087434, 21.5223368904769144365368970892, 23.3214144459651807675198405027, 24.24916133990331503337393072572, 24.97090686328962951674268222680, 25.667308487710781053192410640615, 26.4544377160109221169283447917, 27.69456656692807293195277307539

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