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Properties

Label 1-3724-3724.167-r1-0-0
Degree $1$
Conductor $3724$
Sign $0.204 + 0.978i$
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  + (−0.698 + 0.715i)3-s + (−0.270 − 0.962i)5-s + (−0.0249 − 0.999i)9-s + (0.988 − 0.149i)11-s + (−0.318 − 0.947i)13-s + (0.878 + 0.478i)15-s + (−0.456 + 0.889i)17-s + (0.998 + 0.0498i)23-s + (−0.853 + 0.521i)25-s + (0.733 + 0.680i)27-s + (0.797 + 0.603i)29-s + (0.5 + 0.866i)31-s + (−0.583 + 0.811i)33-s + (0.222 + 0.974i)37-s + (0.900 + 0.433i)39-s + ⋯
L(s)  = 1  + (−0.698 + 0.715i)3-s + (−0.270 − 0.962i)5-s + (−0.0249 − 0.999i)9-s + (0.988 − 0.149i)11-s + (−0.318 − 0.947i)13-s + (0.878 + 0.478i)15-s + (−0.456 + 0.889i)17-s + (0.998 + 0.0498i)23-s + (−0.853 + 0.521i)25-s + (0.733 + 0.680i)27-s + (0.797 + 0.603i)29-s + (0.5 + 0.866i)31-s + (−0.583 + 0.811i)33-s + (0.222 + 0.974i)37-s + (0.900 + 0.433i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3724\)    =    \(2^{2} \cdot 7^{2} \cdot 19\)
Sign: $0.204 + 0.978i$
Analytic conductor: \(400.199\)
Root analytic conductor: \(400.199\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3724} (167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3724,\ (1:\ ),\ 0.204 + 0.978i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.094730207 + 0.8897143168i\)
\(L(\frac12)\) \(\approx\) \(1.094730207 + 0.8897143168i\)
\(L(1)\) \(\approx\) \(0.8668640610 + 0.07425994321i\)
\(L(1)\) \(\approx\) \(0.8668640610 + 0.07425994321i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
19 \( 1 \)
good3 \( 1 + (-0.698 + 0.715i)T \)
5 \( 1 + (-0.270 - 0.962i)T \)
11 \( 1 + (0.988 - 0.149i)T \)
13 \( 1 + (-0.318 - 0.947i)T \)
17 \( 1 + (-0.456 + 0.889i)T \)
23 \( 1 + (0.998 + 0.0498i)T \)
29 \( 1 + (0.797 + 0.603i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.222 + 0.974i)T \)
41 \( 1 + (0.698 - 0.715i)T \)
43 \( 1 + (0.583 - 0.811i)T \)
47 \( 1 + (0.980 + 0.198i)T \)
53 \( 1 + (0.998 + 0.0498i)T \)
59 \( 1 + (0.411 + 0.911i)T \)
61 \( 1 + (-0.921 + 0.388i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (0.921 + 0.388i)T \)
73 \( 1 + (0.318 - 0.947i)T \)
79 \( 1 + (-0.939 + 0.342i)T \)
83 \( 1 + (-0.988 - 0.149i)T \)
89 \( 1 + (-0.853 + 0.521i)T \)
97 \( 1 + (0.766 + 0.642i)T \)
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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

−18.42606434289145039914513046599, −17.55690960558745974205884832463, −17.15225362225466257102880318518, −16.33021930777991826954674147426, −15.64724727724233626392035265715, −14.745075603476550263330759334157, −14.11866238944028409686336383235, −13.61536433299947978350501047308, −12.64340631443828355155398152230, −11.95855885430052088603959997014, −11.271308158328454683212706590658, −11.11055766349207641760683753681, −9.93402171413464860212035307893, −9.33945655590059289000100235297, −8.33560280598590154688604105798, −7.40223439855642795782338796538, −6.94711151651449154648797620815, −6.43176494551592108655559545302, −5.67805794644309775618802311653, −4.55819379183458881466487364657, −4.07762360123562799177002340489, −2.75532779524764401415109899008, −2.28079891385791659574584479046, −1.184121122696542918269712924016, −0.32357371022573541581269163497, 0.82617470169610573213239829175, 1.23730404523449398781969230412, 2.71458003514323956920312023634, 3.70747057103430279898042872691, 4.25658447837608047271737555537, 5.03811998120552943333345986380, 5.60561276683249491779228593153, 6.43319491281039319000145316656, 7.20145103722660950049386490979, 8.32497992776507657332246824634, 8.863134736179985134171160082016, 9.45045622522033154890740817590, 10.42570820843950703702708416583, 10.83149312094987153045244829985, 11.830761404501764575322120188607, 12.26830238936873851579296180834, 12.85815165970039593420306108239, 13.77661656217990002308264577962, 14.71392896701118628320020487333, 15.36703416275431143741335231172, 15.83301717358714454627462202693, 16.67643793653846336832020227626, 17.26614311983159423629318010724, 17.46675030594640230205265795221, 18.51998862453279282751758776659

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