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Properties

Label 1-161-161.10-r0-0-0
Degree $1$
Conductor $161$
Sign $0.999 + 0.00895i$
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  + (−0.786 − 0.618i)2-s + (−0.580 + 0.814i)3-s + (0.235 + 0.971i)4-s + (0.981 − 0.189i)5-s + (0.959 − 0.281i)6-s + (0.415 − 0.909i)8-s + (−0.327 − 0.945i)9-s + (−0.888 − 0.458i)10-s + (0.786 − 0.618i)11-s + (−0.928 − 0.371i)12-s + (−0.841 + 0.540i)13-s + (−0.415 + 0.909i)15-s + (−0.888 + 0.458i)16-s + (0.723 + 0.690i)17-s + (−0.327 + 0.945i)18-s + (0.723 − 0.690i)19-s + ⋯
L(s)  = 1  + (−0.786 − 0.618i)2-s + (−0.580 + 0.814i)3-s + (0.235 + 0.971i)4-s + (0.981 − 0.189i)5-s + (0.959 − 0.281i)6-s + (0.415 − 0.909i)8-s + (−0.327 − 0.945i)9-s + (−0.888 − 0.458i)10-s + (0.786 − 0.618i)11-s + (−0.928 − 0.371i)12-s + (−0.841 + 0.540i)13-s + (−0.415 + 0.909i)15-s + (−0.888 + 0.458i)16-s + (0.723 + 0.690i)17-s + (−0.327 + 0.945i)18-s + (0.723 − 0.690i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $0.999 + 0.00895i$
Analytic conductor: \(0.747680\)
Root analytic conductor: \(0.747680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (0:\ ),\ 0.999 + 0.00895i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7697993161 + 0.003445132471i\)
\(L(\frac12)\) \(\approx\) \(0.7697993161 + 0.003445132471i\)
\(L(1)\) \(\approx\) \(0.7557736072 + 0.02147222656i\)
\(L(1)\) \(\approx\) \(0.7557736072 + 0.02147222656i\)

Euler product

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

−27.85482560332539937932162626476, −26.82180891522304501571827841241, −25.541225288631093328764583279355, −24.93195677055825205854223755516, −24.34979895451784702674127987677, −22.89401287303215843000579792319, −22.41792858802881174043222193069, −20.7308217669564071650124077082, −19.58226484244865821033911236604, −18.61328741678398640086412240441, −17.736525765231946634798193319580, −17.1904471491138062505164120074, −16.248637695810207594905861595980, −14.65653334082465389866336176763, −13.975204797776635270537304837603, −12.60425104341241326759330098806, −11.44970949587793425468063394341, −10.13909789085842049066870351876, −9.419640121770998002570150370427, −7.81571838580260324024618650398, −7.007950104796376873618654064204, −5.947003111719596374557285955861, −5.11450564482107746315335736984, −2.38577193080478920337593418167, −1.18681410069939851579555968539, 1.23642693604633051273336249369, 2.90373393225618260566473153676, 4.27977124005834328814329289439, 5.68903774084679547205746494067, 6.91399234558988972376281776535, 8.682742087376977168771336261681, 9.541204300884214335488762263, 10.22419392366853583820734989698, 11.400643607183732238892456981993, 12.219132552650787111313326626824, 13.59558921465604742696955906428, 14.8972304959986434960177899275, 16.42871558667557616774721407440, 16.94020492002369658698465547610, 17.70274734627538397552150715443, 18.88544973729350468460348824852, 20.05446495313870787002682749042, 21.041525290514505752031677851292, 21.84700313305521645227796193402, 22.28984162952360523162705758583, 24.03903997893606500345028390542, 25.13584457816802888977084976833, 26.24505135344101629995656201561, 26.85847768149697314582289147550, 27.97751260354610979777125248856

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