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Properties

Label 1-1840-1840.1709-r1-0-0
Degree $1$
Conductor $1840$
Sign $0.339 + 0.940i$
Analytic cond. $197.735$
Root an. cond. $197.735$
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  + (−0.909 − 0.415i)3-s + (−0.841 + 0.540i)7-s + (0.654 + 0.755i)9-s + (−0.989 − 0.142i)11-s + (0.540 − 0.841i)13-s + (−0.959 − 0.281i)17-s + (0.281 + 0.959i)19-s + (0.989 − 0.142i)21-s + (−0.281 − 0.959i)27-s + (0.281 − 0.959i)29-s + (0.415 + 0.909i)31-s + (0.841 + 0.540i)33-s + (−0.755 + 0.654i)37-s + (−0.841 + 0.540i)39-s + (0.654 − 0.755i)41-s + ⋯
L(s)  = 1  + (−0.909 − 0.415i)3-s + (−0.841 + 0.540i)7-s + (0.654 + 0.755i)9-s + (−0.989 − 0.142i)11-s + (0.540 − 0.841i)13-s + (−0.959 − 0.281i)17-s + (0.281 + 0.959i)19-s + (0.989 − 0.142i)21-s + (−0.281 − 0.959i)27-s + (0.281 − 0.959i)29-s + (0.415 + 0.909i)31-s + (0.841 + 0.540i)33-s + (−0.755 + 0.654i)37-s + (−0.841 + 0.540i)39-s + (0.654 − 0.755i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.339 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.339 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.339 + 0.940i$
Analytic conductor: \(197.735\)
Root analytic conductor: \(197.735\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (1709, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1840,\ (1:\ ),\ 0.339 + 0.940i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3474290640 + 0.2439893939i\)
\(L(\frac12)\) \(\approx\) \(0.3474290640 + 0.2439893939i\)
\(L(1)\) \(\approx\) \(0.6051932661 - 0.06666413467i\)
\(L(1)\) \(\approx\) \(0.6051932661 - 0.06666413467i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good3 \( 1 + (-0.909 - 0.415i)T \)
7 \( 1 + (-0.841 + 0.540i)T \)
11 \( 1 + (-0.989 - 0.142i)T \)
13 \( 1 + (0.540 - 0.841i)T \)
17 \( 1 + (-0.959 - 0.281i)T \)
19 \( 1 + (0.281 + 0.959i)T \)
29 \( 1 + (0.281 - 0.959i)T \)
31 \( 1 + (0.415 + 0.909i)T \)
37 \( 1 + (-0.755 + 0.654i)T \)
41 \( 1 + (0.654 - 0.755i)T \)
43 \( 1 + (-0.909 - 0.415i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.540 - 0.841i)T \)
59 \( 1 + (0.540 - 0.841i)T \)
61 \( 1 + (0.909 - 0.415i)T \)
67 \( 1 + (0.989 - 0.142i)T \)
71 \( 1 + (0.142 + 0.989i)T \)
73 \( 1 + (-0.959 + 0.281i)T \)
79 \( 1 + (-0.841 - 0.540i)T \)
83 \( 1 + (0.755 - 0.654i)T \)
89 \( 1 + (0.415 - 0.909i)T \)
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

−19.834997727181273983988461109983, −19.0539884785068278275975730164, −18.14547976127531353571466881110, −17.6758978501082685233227737092, −16.75649446408439370849794439031, −16.11759823091387334527461073232, −15.70089956485965599204867884592, −14.85593858291946084824467688789, −13.633480320803021578543125130515, −13.135664321998889025336794637712, −12.4226011680013144043058553354, −11.35069000837105275451186072971, −10.93361394442800451733629854602, −10.10986262671951465347015589841, −9.445054696372075857470806468676, −8.61986932706787364751813643760, −7.3674663519021522469390648225, −6.69913585311832352474940416400, −6.100505969595733627786675409633, −5.054735373912939828994817220702, −4.388147439498109589082220129643, −3.55679282820044378942416266101, −2.51695028875722002954854790640, −1.21056309532422736094482612390, −0.15806858603980732012187360429, 0.54876536920347453945827764060, 1.808374933870624989152984747980, 2.75660548611822317397709694534, 3.67569891051143023865557552546, 4.946825144644522051094436158205, 5.52452663656828784796265143047, 6.295288293681309980188333211581, 6.928616335702687715577148681, 7.994723359753240256666635011150, 8.57659819962232869174065656671, 9.90922224939427481535849967327, 10.28541396986845906488085207927, 11.21809192241358094474341764146, 11.92646175520879477345181876428, 12.78619416915473542934381790430, 13.144817056500482491190631708984, 13.95311923996734252933384440201, 15.25759657210030751520774246557, 15.98462444667083314338780844632, 16.11922000816825184159004570697, 17.4266933321904453076473084736, 17.805895110820953154727328825498, 18.75767991906306255837576963919, 18.9847984893985960096730977798, 20.09638561119355410585970277436

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