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Properties

Label 1-3724-3724.787-r1-0-0
Degree $1$
Conductor $3724$
Sign $-0.931 + 0.363i$
Analytic cond. $400.199$
Root an. cond. $400.199$
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  + (−0.365 − 0.930i)3-s + (−0.623 + 0.781i)5-s + (−0.733 + 0.680i)9-s + (−0.955 + 0.294i)11-s + (−0.733 − 0.680i)13-s + (0.955 + 0.294i)15-s + (−0.0747 + 0.997i)17-s + (−0.0747 − 0.997i)23-s + (−0.222 − 0.974i)25-s + (0.900 + 0.433i)27-s + (−0.0747 + 0.997i)29-s + (0.5 + 0.866i)31-s + (0.623 + 0.781i)33-s + (−0.0747 + 0.997i)37-s + (−0.365 + 0.930i)39-s + ⋯
L(s)  = 1  + (−0.365 − 0.930i)3-s + (−0.623 + 0.781i)5-s + (−0.733 + 0.680i)9-s + (−0.955 + 0.294i)11-s + (−0.733 − 0.680i)13-s + (0.955 + 0.294i)15-s + (−0.0747 + 0.997i)17-s + (−0.0747 − 0.997i)23-s + (−0.222 − 0.974i)25-s + (0.900 + 0.433i)27-s + (−0.0747 + 0.997i)29-s + (0.5 + 0.866i)31-s + (0.623 + 0.781i)33-s + (−0.0747 + 0.997i)37-s + (−0.365 + 0.930i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.931 + 0.363i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.931 + 0.363i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3724\)    =    \(2^{2} \cdot 7^{2} \cdot 19\)
Sign: $-0.931 + 0.363i$
Analytic conductor: \(400.199\)
Root analytic conductor: \(400.199\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3724} (787, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3724,\ (1:\ ),\ -0.931 + 0.363i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08167668470 + 0.4337145405i\)
\(L(\frac12)\) \(\approx\) \(0.08167668470 + 0.4337145405i\)
\(L(1)\) \(\approx\) \(0.6734138870 + 0.002436031149i\)
\(L(1)\) \(\approx\) \(0.6734138870 + 0.002436031149i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
19 \( 1 \)
good3 \( 1 + (-0.365 - 0.930i)T \)
5 \( 1 + (-0.623 + 0.781i)T \)
11 \( 1 + (-0.955 + 0.294i)T \)
13 \( 1 + (-0.733 - 0.680i)T \)
17 \( 1 + (-0.0747 + 0.997i)T \)
23 \( 1 + (-0.0747 - 0.997i)T \)
29 \( 1 + (-0.0747 + 0.997i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (-0.0747 + 0.997i)T \)
41 \( 1 + (-0.988 - 0.149i)T \)
43 \( 1 + (-0.365 + 0.930i)T \)
47 \( 1 + (0.955 - 0.294i)T \)
53 \( 1 + (0.900 + 0.433i)T \)
59 \( 1 + (0.988 - 0.149i)T \)
61 \( 1 + (-0.826 - 0.563i)T \)
67 \( 1 + T \)
71 \( 1 + (0.0747 + 0.997i)T \)
73 \( 1 + (-0.955 - 0.294i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.222 - 0.974i)T \)
89 \( 1 + (-0.733 + 0.680i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−18.03977359849572957863306553303, −17.25352365812143309330861341485, −16.66036967022968976822136241077, −16.14643819592497271205111511831, −15.417631388524434927909056314462, −15.13117668867408942310572536197, −13.94444124125208161816358251964, −13.442116336642188370795624326190, −12.433328101154832074781453999917, −11.71400621288899243950514721560, −11.430396590983984179022254947063, −10.45692849572943964198725469265, −9.69413400633926762196091053149, −9.20448391596816427043906493244, −8.39592516014713698123194621914, −7.64309535748037307972165336822, −6.89590512257094591639232007237, −5.62334432177896647743029983802, −5.31374984568053115952718506340, −4.460275111042342750479519964774, −3.9408234073841342769310139622, −2.98868673192199591024962919249, −2.096802044574462691326317388676, −0.6465947707741747637450990702, −0.12603614588668507227240144258, 0.840490925600927388084471497418, 1.975208301279476714956654668525, 2.71196945459950329850317425314, 3.33248745265494050958894811394, 4.5576122512872509531107299419, 5.2413933401758512426088091875, 6.117899043054908573245474274983, 6.88027188601126980795841692478, 7.338563897511155702430037422665, 8.21866084816498418690809217415, 8.489188677699997650007227925000, 10.08203105716659105961732052188, 10.46935871620672597184860341623, 11.0980478642181564690691555027, 12.030528372452900067264253725058, 12.474116927177065755157243708224, 13.09056060765397202532137735577, 13.93272454544338612269387214504, 14.71247297455328273946635295529, 15.208432047155145655441696063912, 16.01033381694713700849635396752, 16.856452566249688464875850697474, 17.51589515025011366998542432630, 18.24065677769755909682344361646, 18.59696913781179644216217265213

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