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Properties

Label 1-161-161.75-r1-0-0
Degree $1$
Conductor $161$
Sign $0.925 - 0.377i$
Analytic cond. $17.3018$
Root an. cond. $17.3018$
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  + (−0.327 + 0.945i)2-s + (0.888 − 0.458i)3-s + (−0.786 − 0.618i)4-s + (0.995 − 0.0950i)5-s + (0.142 + 0.989i)6-s + (0.841 − 0.540i)8-s + (0.580 − 0.814i)9-s + (−0.235 + 0.971i)10-s + (−0.327 − 0.945i)11-s + (−0.981 − 0.189i)12-s + (0.959 − 0.281i)13-s + (0.841 − 0.540i)15-s + (0.235 + 0.971i)16-s + (−0.928 − 0.371i)17-s + (0.580 + 0.814i)18-s + (−0.928 + 0.371i)19-s + ⋯
L(s)  = 1  + (−0.327 + 0.945i)2-s + (0.888 − 0.458i)3-s + (−0.786 − 0.618i)4-s + (0.995 − 0.0950i)5-s + (0.142 + 0.989i)6-s + (0.841 − 0.540i)8-s + (0.580 − 0.814i)9-s + (−0.235 + 0.971i)10-s + (−0.327 − 0.945i)11-s + (−0.981 − 0.189i)12-s + (0.959 − 0.281i)13-s + (0.841 − 0.540i)15-s + (0.235 + 0.971i)16-s + (−0.928 − 0.371i)17-s + (0.580 + 0.814i)18-s + (−0.928 + 0.371i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.925 - 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.925 - 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $0.925 - 0.377i$
Analytic conductor: \(17.3018\)
Root analytic conductor: \(17.3018\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (1:\ ),\ 0.925 - 0.377i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.194561848 - 0.4303590820i\)
\(L(\frac12)\) \(\approx\) \(2.194561848 - 0.4303590820i\)
\(L(1)\) \(\approx\) \(1.366742866 + 0.06361338761i\)
\(L(1)\) \(\approx\) \(1.366742866 + 0.06361338761i\)

Euler product

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

−27.76742114561423112663302213133, −26.54470413112769149328356701152, −25.81543370143685953282653074160, −25.27007470045607485934310049302, −23.5941144089408362613063062665, −22.20284752046547827774416137580, −21.53825643974675082254099708777, −20.6743023320985700685501624362, −20.01731897497909259036647023197, −18.8215138503484749327770465471, −17.95401487949372260072358622593, −16.948185188821004291451248459619, −15.54174752017007206682524316576, −14.311493737280139388625773619968, −13.39616583582339278752983601976, −12.67465679105164292762283747153, −10.88073653558790040400908723219, −10.25451854007739544392928746399, −9.14067241113582006012490497643, −8.51940038091905773393736491009, −6.922073442388614426792326506642, −5.01144425649703293276925730410, −3.887147736844183825273050641164, −2.503068686905203478050959254489, −1.68064778285336542020689169152, 0.872450097333271644514182337832, 2.36514253199719471603713379648, 4.10130956978470114169890853055, 5.80662194402742684123303772930, 6.508460356431363979048117497389, 7.9363911274195998976106539664, 8.746147004648952124028646498737, 9.61282758843428311030118993861, 10.87056259843541183253070308919, 13.07605977055917046188979287067, 13.45616879373701723656136885995, 14.42477508914161508381272118858, 15.46048027940634085407194866048, 16.5370800010079475097213491314, 17.73134345129012355501076000402, 18.4405617189153537005390873040, 19.31269988513483035772982812269, 20.622552412084274637899060967625, 21.55268575111892134773187861156, 22.878095673582057803616574622587, 24.062632753738250831026226314754, 24.71321447012081713039845998091, 25.55561394929292306391116990971, 26.212354423764932520732297524593, 27.10422966700017306290296251212

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