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Properties

Label 1-19e2-361.100-r0-0-0
Degree $1$
Conductor $361$
Sign $-0.830 + 0.557i$
Analytic cond. $1.67647$
Root an. cond. $1.67647$
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  + (−0.649 + 0.760i)2-s + (−0.800 + 0.599i)3-s + (−0.155 − 0.987i)4-s + (0.741 + 0.670i)5-s + (0.0642 − 0.997i)6-s + (−0.754 + 0.656i)7-s + (0.851 + 0.523i)8-s + (0.280 − 0.959i)9-s + (−0.991 + 0.128i)10-s + (0.993 − 0.110i)11-s + (0.716 + 0.697i)12-s + (−0.227 + 0.973i)13-s + (−0.00918 − 0.999i)14-s + (−0.995 − 0.0917i)15-s + (−0.951 + 0.307i)16-s + (0.933 − 0.359i)17-s + ⋯
L(s)  = 1  + (−0.649 + 0.760i)2-s + (−0.800 + 0.599i)3-s + (−0.155 − 0.987i)4-s + (0.741 + 0.670i)5-s + (0.0642 − 0.997i)6-s + (−0.754 + 0.656i)7-s + (0.851 + 0.523i)8-s + (0.280 − 0.959i)9-s + (−0.991 + 0.128i)10-s + (0.993 − 0.110i)11-s + (0.716 + 0.697i)12-s + (−0.227 + 0.973i)13-s + (−0.00918 − 0.999i)14-s + (−0.995 − 0.0917i)15-s + (−0.951 + 0.307i)16-s + (0.933 − 0.359i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.830 + 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.830 + 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(361\)    =    \(19^{2}\)
Sign: $-0.830 + 0.557i$
Analytic conductor: \(1.67647\)
Root analytic conductor: \(1.67647\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{361} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 361,\ (0:\ ),\ -0.830 + 0.557i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2177975520 + 0.7157239110i\)
\(L(\frac12)\) \(\approx\) \(0.2177975520 + 0.7157239110i\)
\(L(1)\) \(\approx\) \(0.5127180208 + 0.4638568092i\)
\(L(1)\) \(\approx\) \(0.5127180208 + 0.4638568092i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
good2 \( 1 + (-0.649 + 0.760i)T \)
3 \( 1 + (-0.800 + 0.599i)T \)
5 \( 1 + (0.741 + 0.670i)T \)
7 \( 1 + (-0.754 + 0.656i)T \)
11 \( 1 + (0.993 - 0.110i)T \)
13 \( 1 + (-0.227 + 0.973i)T \)
17 \( 1 + (0.933 - 0.359i)T \)
23 \( 1 + (0.919 - 0.393i)T \)
29 \( 1 + (0.515 + 0.856i)T \)
31 \( 1 + (0.635 - 0.771i)T \)
37 \( 1 + (-0.401 + 0.915i)T \)
41 \( 1 + (-0.263 + 0.964i)T \)
43 \( 1 + (0.606 - 0.794i)T \)
47 \( 1 + (-0.896 + 0.443i)T \)
53 \( 1 + (0.832 + 0.554i)T \)
59 \( 1 + (-0.703 - 0.710i)T \)
61 \( 1 + (-0.621 + 0.783i)T \)
67 \( 1 + (-0.979 - 0.200i)T \)
71 \( 1 + (-0.621 - 0.783i)T \)
73 \( 1 + (-0.777 - 0.628i)T \)
79 \( 1 + (0.384 + 0.922i)T \)
83 \( 1 + (-0.926 + 0.376i)T \)
89 \( 1 + (-0.777 + 0.628i)T \)
97 \( 1 + (-0.979 + 0.200i)T \)
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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

−24.70715774897787891011443760429, −23.16561743163758061094169200219, −22.64819370583768020439668534963, −21.65216932842710035903571482015, −20.82023547681053670348689019858, −19.63625184399129172091194413497, −19.31989560295302349520273899643, −17.996340321700447513046660485558, −17.25098702560001651901774444719, −16.901552913869785418187434208383, −15.95236615537120221268935639984, −14.044416028023331767518311887949, −13.125862081914056388040426749474, −12.553887761349856923989366246383, −11.76192095537711849717292257298, −10.46815325185071488544894650525, −9.97343737232359178517062982750, −8.8778728285683156000922646315, −7.69472155251291668890224623175, −6.73569845286534598069366809635, −5.64180259213130670890354690233, −4.37618104950339071767051645752, −3.021930634389461935800071967917, −1.549206187355741645837784179, −0.74311729341781206545819895262, 1.36320524007075621204154888590, 3.04386445817755567434873418050, 4.61939930994970147246956342437, 5.71868001008749342898562344282, 6.44516983208814565529855221793, 7.026654769047377045169061159732, 8.87804604482611107272396589643, 9.522905096912572159282976832060, 10.168385627531419209436872385467, 11.25291221024432586311690166820, 12.19483545310735959401470386510, 13.716732429072369321058477567143, 14.65889264448401241972193782310, 15.32043054570812489996344114933, 16.611204352678447052581124037216, 16.76267245752310853877998909229, 17.90468851165727617139297383397, 18.70499060008218711410224044826, 19.35318303971393170979137228031, 20.86989226063392541853787139805, 21.866584485336723792595675791294, 22.50163421377452451721429239362, 23.20765055675059135223806564973, 24.36600096172058731104898208375, 25.26080543732826504623482441308

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