Properties

Label 1-1840-1840.123-r0-0-0
Degree $1$
Conductor $1840$
Sign $-0.0526 - 0.998i$
Analytic cond. $8.54492$
Root an. cond. $8.54492$
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  + (0.654 − 0.755i)3-s + (0.909 − 0.415i)7-s + (−0.142 − 0.989i)9-s + (0.281 + 0.959i)11-s + (0.415 − 0.909i)13-s + (−0.540 − 0.841i)17-s + (0.540 − 0.841i)19-s + (0.281 − 0.959i)21-s + (−0.841 − 0.540i)27-s + (0.540 + 0.841i)29-s + (0.654 + 0.755i)31-s + (0.909 + 0.415i)33-s + (−0.142 − 0.989i)37-s + (−0.415 − 0.909i)39-s + (0.142 − 0.989i)41-s + ⋯
L(s)  = 1  + (0.654 − 0.755i)3-s + (0.909 − 0.415i)7-s + (−0.142 − 0.989i)9-s + (0.281 + 0.959i)11-s + (0.415 − 0.909i)13-s + (−0.540 − 0.841i)17-s + (0.540 − 0.841i)19-s + (0.281 − 0.959i)21-s + (−0.841 − 0.540i)27-s + (0.540 + 0.841i)29-s + (0.654 + 0.755i)31-s + (0.909 + 0.415i)33-s + (−0.142 − 0.989i)37-s + (−0.415 − 0.909i)39-s + (0.142 − 0.989i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0526 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0526 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $-0.0526 - 0.998i$
Analytic conductor: \(8.54492\)
Root analytic conductor: \(8.54492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1840,\ (0:\ ),\ -0.0526 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.622195147 - 1.710017740i\)
\(L(\frac12)\) \(\approx\) \(1.622195147 - 1.710017740i\)
\(L(1)\) \(\approx\) \(1.373693686 - 0.5980088302i\)
\(L(1)\) \(\approx\) \(1.373693686 - 0.5980088302i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good3 \( 1 + (-0.654 + 0.755i)T \)
7 \( 1 + (-0.909 + 0.415i)T \)
11 \( 1 + (-0.281 - 0.959i)T \)
13 \( 1 + (-0.415 + 0.909i)T \)
17 \( 1 + (0.540 + 0.841i)T \)
19 \( 1 + (-0.540 + 0.841i)T \)
29 \( 1 + (-0.540 - 0.841i)T \)
31 \( 1 + (-0.654 - 0.755i)T \)
37 \( 1 + (0.142 + 0.989i)T \)
41 \( 1 + (-0.142 + 0.989i)T \)
43 \( 1 + (0.654 - 0.755i)T \)
47 \( 1 + iT \)
53 \( 1 + (0.415 + 0.909i)T \)
59 \( 1 + (-0.909 - 0.415i)T \)
61 \( 1 + (0.755 - 0.654i)T \)
67 \( 1 + (0.959 + 0.281i)T \)
71 \( 1 + (0.959 + 0.281i)T \)
73 \( 1 + (-0.540 + 0.841i)T \)
79 \( 1 + (-0.415 + 0.909i)T \)
83 \( 1 + (-0.142 - 0.989i)T \)
89 \( 1 + (0.654 - 0.755i)T \)
97 \( 1 + (-0.989 - 0.142i)T \)
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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

−20.5143128731558945593175259079, −19.6133959816366104911282642361, −18.8924900984061407005584910507, −18.36352728380089972356022314846, −17.13778040464457794963137937187, −16.71207125335263618245231021471, −15.75234216258739560683243379134, −15.22034462267096977827100602726, −14.3815650493344035386535622666, −13.88006303872007265068652493388, −13.19451326080473573656783802886, −11.782822864464026535665024929115, −11.4716597561522427705269057269, −10.54174091378965081674175198988, −9.79518574857339270514515102587, −8.82409182846968641523648674560, −8.411619927905020722948475942960, −7.753494273746167890145056098832, −6.422041907912800685376756634960, −5.70304572696088836854939012112, −4.68134969019696461582759312601, −4.0553526351410021494442744380, −3.20771541811362292578344332844, −2.180475069451016381554067888198, −1.37283030865889820074041596079, 0.799090966591538168419447142827, 1.61573279260291244783446431576, 2.560038779359571230070102036167, 3.38069346984337196887962886395, 4.4941063358160737228033770994, 5.17376820663961663421189087298, 6.39440461506136260282985623658, 7.222888526900057168692347862221, 7.62889681900578898999395349082, 8.59910311855993375307473006694, 9.17554695673312557133703442307, 10.18579795898342448753113398003, 11.050688223451993402316296695856, 11.86867025677864071133458417086, 12.565972214601366171842689632283, 13.401888101408907762106463915122, 13.98306343554256619536413406686, 14.69315794424378823496285198834, 15.37004343554031408228712341027, 16.1613903476244044164732124564, 17.46918650693483960450681196772, 17.90685843149873664291698607403, 18.13701791528832139684748867233, 19.541399463039808001693358714003, 19.817603339633353665780966121535

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