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Properties

Label 1-3724-3724.151-r0-0-0
Degree $1$
Conductor $3724$
Sign $0.996 + 0.0853i$
Analytic cond. $17.2941$
Root an. cond. $17.2941$
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  + (0.0747 + 0.997i)3-s + (0.826 − 0.563i)5-s + (−0.988 + 0.149i)9-s + (0.988 + 0.149i)11-s + (−0.623 + 0.781i)13-s + (0.623 + 0.781i)15-s + (0.955 − 0.294i)17-s + (−0.955 − 0.294i)23-s + (0.365 − 0.930i)25-s + (−0.222 − 0.974i)27-s + (0.222 − 0.974i)29-s + (−0.5 − 0.866i)31-s + (−0.0747 + 0.997i)33-s + (0.733 + 0.680i)37-s + (−0.826 − 0.563i)39-s + ⋯
L(s)  = 1  + (0.0747 + 0.997i)3-s + (0.826 − 0.563i)5-s + (−0.988 + 0.149i)9-s + (0.988 + 0.149i)11-s + (−0.623 + 0.781i)13-s + (0.623 + 0.781i)15-s + (0.955 − 0.294i)17-s + (−0.955 − 0.294i)23-s + (0.365 − 0.930i)25-s + (−0.222 − 0.974i)27-s + (0.222 − 0.974i)29-s + (−0.5 − 0.866i)31-s + (−0.0747 + 0.997i)33-s + (0.733 + 0.680i)37-s + (−0.826 − 0.563i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3724\)    =    \(2^{2} \cdot 7^{2} \cdot 19\)
Sign: $0.996 + 0.0853i$
Analytic conductor: \(17.2941\)
Root analytic conductor: \(17.2941\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3724} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3724,\ (0:\ ),\ 0.996 + 0.0853i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.125264168 + 0.09089501039i\)
\(L(\frac12)\) \(\approx\) \(2.125264168 + 0.09089501039i\)
\(L(1)\) \(\approx\) \(1.281295611 + 0.2058026351i\)
\(L(1)\) \(\approx\) \(1.281295611 + 0.2058026351i\)

Euler product

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

−18.51500490170284445101019771101, −17.85530317291683190978461140864, −17.51148289751741962081208952711, −16.782475709160149139235523395603, −15.99761685304751300239113524759, −14.73153463160598690263786236647, −14.42000921919095633838944972802, −13.99927021901091904035339459647, −12.981130819714368925060732171351, −12.53984244091987651794159558905, −11.839232319660891110217063244131, −10.99212181217754954017674517981, −10.29821646653623912713490873622, −9.48404810992367549663557791299, −8.84272536299269955576278152092, −7.87987067448615102535810754189, −7.29133290312562763310651920904, −6.62265693243463311975930421021, −5.73744142723231502459831433154, −5.54159996784533214312912716019, −4.105847807163545825381603761682, −3.13920984986291505861735335420, −2.57513051706030868125887755710, −1.61169146718380647620188699802, −0.999726535834168758844938542131, 0.675329268509991681267171987601, 1.88055233672183046183593492923, 2.5218704696160385232302688405, 3.64751359469152464126272854814, 4.34674009560750510735667862751, 4.89680916226693190791382137937, 5.87017875983916206874907563148, 6.25150823003294831218583393661, 7.421757188572526140489883239638, 8.302494089589412940189243679407, 9.06669474028869443024394821681, 9.78135936109959204212051016576, 9.83019616665459753886227271083, 10.930469728359678855164334059402, 11.87730663348935433269138242947, 12.13100783906007977178811482078, 13.31545020007539724003268158188, 13.98627365368026405511937644394, 14.523618010312449814801552344207, 15.072165793415183506415974141, 16.16151426004056554886691566548, 16.61786259187216486495972525852, 17.02880159957437394783211953319, 17.74495586641591007956791964182, 18.607849899791141661363105013406

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