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Properties

Label 1-3724-3724.1251-r0-0-0
Degree $1$
Conductor $3724$
Sign $-0.562 - 0.826i$
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  + (0.270 − 0.962i)3-s + (−0.270 + 0.962i)5-s + (−0.853 − 0.521i)9-s + (−0.623 + 0.781i)11-s + (−0.980 + 0.198i)13-s + (0.853 + 0.521i)15-s + (−0.542 + 0.840i)17-s + (−0.456 + 0.889i)23-s + (−0.853 − 0.521i)25-s + (−0.733 + 0.680i)27-s + (0.921 + 0.388i)29-s + (−0.5 − 0.866i)31-s + (0.583 + 0.811i)33-s + (0.955 − 0.294i)37-s + (−0.0747 + 0.997i)39-s + ⋯
L(s)  = 1  + (0.270 − 0.962i)3-s + (−0.270 + 0.962i)5-s + (−0.853 − 0.521i)9-s + (−0.623 + 0.781i)11-s + (−0.980 + 0.198i)13-s + (0.853 + 0.521i)15-s + (−0.542 + 0.840i)17-s + (−0.456 + 0.889i)23-s + (−0.853 − 0.521i)25-s + (−0.733 + 0.680i)27-s + (0.921 + 0.388i)29-s + (−0.5 − 0.866i)31-s + (0.583 + 0.811i)33-s + (0.955 − 0.294i)37-s + (−0.0747 + 0.997i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.562 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.562 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3724\)    =    \(2^{2} \cdot 7^{2} \cdot 19\)
Sign: $-0.562 - 0.826i$
Analytic conductor: \(17.2941\)
Root analytic conductor: \(17.2941\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3724} (1251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3724,\ (0:\ ),\ -0.562 - 0.826i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2086577522 - 0.3943675112i\)
\(L(\frac12)\) \(\approx\) \(0.2086577522 - 0.3943675112i\)
\(L(1)\) \(\approx\) \(0.8094095413 - 0.06825435609i\)
\(L(1)\) \(\approx\) \(0.8094095413 - 0.06825435609i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
19 \( 1 \)
good3 \( 1 + (0.270 - 0.962i)T \)
5 \( 1 + (-0.270 + 0.962i)T \)
11 \( 1 + (-0.623 + 0.781i)T \)
13 \( 1 + (-0.980 + 0.198i)T \)
17 \( 1 + (-0.542 + 0.840i)T \)
23 \( 1 + (-0.456 + 0.889i)T \)
29 \( 1 + (0.921 + 0.388i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.955 - 0.294i)T \)
41 \( 1 + (0.969 - 0.246i)T \)
43 \( 1 + (-0.995 + 0.0995i)T \)
47 \( 1 + (-0.318 + 0.947i)T \)
53 \( 1 + (-0.998 + 0.0498i)T \)
59 \( 1 + (0.995 - 0.0995i)T \)
61 \( 1 + (0.797 - 0.603i)T \)
67 \( 1 + (-0.766 - 0.642i)T \)
71 \( 1 + (0.124 - 0.992i)T \)
73 \( 1 + (0.661 - 0.749i)T \)
79 \( 1 + (0.939 + 0.342i)T \)
83 \( 1 + (-0.988 + 0.149i)T \)
89 \( 1 + (-0.878 + 0.478i)T \)
97 \( 1 + (-0.173 - 0.984i)T \)
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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

−19.03893924685654160884984983555, −18.070942929437947541866501493504, −17.36038452967527727962168607526, −16.50329624236009122656799335947, −16.19403534459190180978102012721, −15.62674791958610505238686823189, −14.7958093811788893334101210018, −14.14463608695699990382931221657, −13.377267211345729675996354030843, −12.72059853824436907323325678975, −11.8093959839570346522268218062, −11.29833411227907550428721281474, −10.35103683326170957362507217591, −9.81998718336374760647287959769, −9.051819755797086857236752951148, −8.38022357078822140958732919780, −7.9479866211128024167355348044, −6.88622873158510553283002077618, −5.78676080930773737691744055739, −5.08345942904978150458001054858, −4.63510748044873972170904812280, −3.85541055355734932761493599291, −2.86480654775827959452821383706, −2.31393709800701050355842836966, −0.83495832676324541670092618929, 0.14449257362843803121890756796, 1.6594267224701922045016831560, 2.28246841198369505001560533813, 2.91784330495814994818526329405, 3.83864570541530983691797934866, 4.74058498881856378086564219924, 5.79940191349932686928648925761, 6.4809273201729982610884604816, 7.167792250174665643645880264142, 7.731805946538797756918007157915, 8.23390993959796829572682441224, 9.421955425018046795328092513801, 9.9307015403374500071352956357, 10.93575827097511268071969401008, 11.43519858487543852007009764412, 12.37057679182852135475224937140, 12.7370764258676395641907651565, 13.63022142870528451934710726272, 14.27550539624436255244062304320, 14.96672702630412801539782605586, 15.32708725391350058023252907129, 16.36162118744896376634473496505, 17.35608715057263395580020479573, 17.85982228140244045485066167797, 18.26432163050137931151100179395

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