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Properties

Label 1-3724-3724.531-r1-0-0
Degree $1$
Conductor $3724$
Sign $0.545 - 0.838i$
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  + (−0.623 + 0.781i)3-s + (−0.623 + 0.781i)5-s + (−0.222 − 0.974i)9-s + (0.222 − 0.974i)11-s + (−0.222 + 0.974i)13-s + (−0.222 − 0.974i)15-s + (0.900 − 0.433i)17-s + (0.900 + 0.433i)23-s + (−0.222 − 0.974i)25-s + (0.900 + 0.433i)27-s + (0.900 − 0.433i)29-s − 31-s + (0.623 + 0.781i)33-s + (0.900 − 0.433i)37-s + (−0.623 − 0.781i)39-s + ⋯
L(s)  = 1  + (−0.623 + 0.781i)3-s + (−0.623 + 0.781i)5-s + (−0.222 − 0.974i)9-s + (0.222 − 0.974i)11-s + (−0.222 + 0.974i)13-s + (−0.222 − 0.974i)15-s + (0.900 − 0.433i)17-s + (0.900 + 0.433i)23-s + (−0.222 − 0.974i)25-s + (0.900 + 0.433i)27-s + (0.900 − 0.433i)29-s − 31-s + (0.623 + 0.781i)33-s + (0.900 − 0.433i)37-s + (−0.623 − 0.781i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.545 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.545 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3724\)    =    \(2^{2} \cdot 7^{2} \cdot 19\)
Sign: $0.545 - 0.838i$
Analytic conductor: \(400.199\)
Root analytic conductor: \(400.199\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3724} (531, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3724,\ (1:\ ),\ 0.545 - 0.838i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8581239957 - 0.4653298432i\)
\(L(\frac12)\) \(\approx\) \(0.8581239957 - 0.4653298432i\)
\(L(1)\) \(\approx\) \(0.7690512668 + 0.1852286674i\)
\(L(1)\) \(\approx\) \(0.7690512668 + 0.1852286674i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
19 \( 1 \)
good3 \( 1 + (-0.623 + 0.781i)T \)
5 \( 1 + (-0.623 + 0.781i)T \)
11 \( 1 + (0.222 - 0.974i)T \)
13 \( 1 + (-0.222 + 0.974i)T \)
17 \( 1 + (0.900 - 0.433i)T \)
23 \( 1 + (0.900 + 0.433i)T \)
29 \( 1 + (0.900 - 0.433i)T \)
31 \( 1 - T \)
37 \( 1 + (0.900 - 0.433i)T \)
41 \( 1 + (0.623 - 0.781i)T \)
43 \( 1 + (-0.623 - 0.781i)T \)
47 \( 1 + (-0.222 + 0.974i)T \)
53 \( 1 + (0.900 + 0.433i)T \)
59 \( 1 + (-0.623 - 0.781i)T \)
61 \( 1 + (0.900 - 0.433i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.900 - 0.433i)T \)
73 \( 1 + (0.222 + 0.974i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.222 - 0.974i)T \)
89 \( 1 + (-0.222 - 0.974i)T \)
97 \( 1 + T \)
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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

−18.35116358788650280002517355230, −17.97986213751842700622169458815, −17.0949829006889893780412593910, −16.67135804184066246761451842830, −16.04191645952817918476080655402, −14.96710144454528448827386978463, −14.70028065816128024183177072742, −13.40211116665835374456826676440, −12.874129572024877957891082557801, −12.39415766559070486112059615240, −11.86113269791160303335725562653, −11.06007427580260074996013286990, −10.29507989311305895003898008131, −9.52717601685135149789943103106, −8.484907907897582898975567231820, −7.96322790813138922849771777578, −7.30222537682212513917137489139, −6.6249789228373092788628465501, −5.6215235861495190825517246013, −5.06172992513279340027153530246, −4.39777664685875622441812822888, −3.34297624420470019042681901212, −2.39173412358172366791381989105, −1.297676889382376380463965696983, −0.84652417089153842209419427319, 0.22888033213146277437528915259, 1.04605091866451532853546927165, 2.4711029105958639069499114819, 3.346616222360563944406226078, 3.83507622135647922553572869152, 4.67273096706761534751136364670, 5.51430836324982497457705392557, 6.237083654343997227919568175273, 6.94867282916766247839264257851, 7.66142850971313142230187345401, 8.67379027400530530460131935338, 9.34341537046589989381520236279, 10.0590826333206644221152601432, 10.88842543954330404170650733523, 11.32663720948893330960735708557, 11.85999743392234619861811393542, 12.59301265536572404341823859779, 13.793063367675851907058683311378, 14.38068390858906668471107917980, 14.894443660697362673991475831285, 15.81133233538620809481787003650, 16.19503664465861297249019328633, 16.879303405226965028350622715095, 17.53122256879745335894148175994, 18.51046407815984587992167402638

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