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Properties

Label 1-2541-2541.32-r0-0-0
Degree $1$
Conductor $2541$
Sign $0.669 + 0.742i$
Analytic cond. $11.8003$
Root an. cond. $11.8003$
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  + (0.723 − 0.690i)2-s + (0.0475 − 0.998i)4-s + (0.327 + 0.945i)5-s + (−0.654 − 0.755i)8-s + (0.888 + 0.458i)10-s + (−0.841 − 0.540i)13-s + (−0.995 − 0.0950i)16-s + (−0.786 + 0.618i)17-s + (0.786 + 0.618i)19-s + (0.959 − 0.281i)20-s + (0.995 + 0.0950i)23-s + (−0.786 + 0.618i)25-s + (−0.981 + 0.189i)26-s + (−0.142 + 0.989i)29-s + (−0.888 − 0.458i)31-s + (−0.786 + 0.618i)32-s + ⋯
L(s)  = 1  + (0.723 − 0.690i)2-s + (0.0475 − 0.998i)4-s + (0.327 + 0.945i)5-s + (−0.654 − 0.755i)8-s + (0.888 + 0.458i)10-s + (−0.841 − 0.540i)13-s + (−0.995 − 0.0950i)16-s + (−0.786 + 0.618i)17-s + (0.786 + 0.618i)19-s + (0.959 − 0.281i)20-s + (0.995 + 0.0950i)23-s + (−0.786 + 0.618i)25-s + (−0.981 + 0.189i)26-s + (−0.142 + 0.989i)29-s + (−0.888 − 0.458i)31-s + (−0.786 + 0.618i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.669 + 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.669 + 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2541\)    =    \(3 \cdot 7 \cdot 11^{2}\)
Sign: $0.669 + 0.742i$
Analytic conductor: \(11.8003\)
Root analytic conductor: \(11.8003\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2541} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2541,\ (0:\ ),\ 0.669 + 0.742i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.574153782 + 0.6998565014i\)
\(L(\frac12)\) \(\approx\) \(1.574153782 + 0.6998565014i\)
\(L(1)\) \(\approx\) \(1.358215101 - 0.2132472759i\)
\(L(1)\) \(\approx\) \(1.358215101 - 0.2132472759i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.723 - 0.690i)T \)
5 \( 1 + (0.327 + 0.945i)T \)
13 \( 1 + (-0.841 - 0.540i)T \)
17 \( 1 + (-0.786 + 0.618i)T \)
19 \( 1 + (0.786 + 0.618i)T \)
23 \( 1 + (0.995 + 0.0950i)T \)
29 \( 1 + (-0.142 + 0.989i)T \)
31 \( 1 + (-0.888 - 0.458i)T \)
37 \( 1 + (0.0475 + 0.998i)T \)
41 \( 1 + (-0.959 - 0.281i)T \)
43 \( 1 + (0.654 + 0.755i)T \)
47 \( 1 + (-0.235 + 0.971i)T \)
53 \( 1 + (0.995 - 0.0950i)T \)
59 \( 1 + (-0.723 - 0.690i)T \)
61 \( 1 + (-0.235 + 0.971i)T \)
67 \( 1 + (0.235 + 0.971i)T \)
71 \( 1 + (0.142 - 0.989i)T \)
73 \( 1 + (-0.580 + 0.814i)T \)
79 \( 1 + (0.327 + 0.945i)T \)
83 \( 1 + (0.415 - 0.909i)T \)
89 \( 1 + (-0.928 - 0.371i)T \)
97 \( 1 + (-0.654 - 0.755i)T \)
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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

−19.55476917068525540560380709752, −18.35656184390228517588784505036, −17.68025238803887110011924546853, −16.96586665241418338821016292392, −16.53349993071632576657789977171, −15.71748226299039856137945317567, −15.14881085689099501426577871815, −14.22947960210291899799693990443, −13.57019310560402710314836692586, −13.09955803441810772376836837323, −12.21469522725450212038842788852, −11.74494104307897355528050271647, −10.82559956173722870432356859086, −9.4643639031942957805827032451, −9.14795647642502704714131478589, −8.32102816654338255013109602912, −7.30952703198040836108768127543, −6.89665157575345693806341993508, −5.81611341224322553028265240348, −5.09150676781060422174202177243, −4.65062496314784604437499450473, −3.75586439423543027636634734411, −2.667065220047709108285820527443, −1.90598122505906145189241432217, −0.402958414277636353987853777124, 1.2527190234955231364567149920, 2.12669999156355970395363687976, 2.96241304472704858761000860383, 3.48998144849660928789691824196, 4.53930728557501151817345736253, 5.36755001470121509509950314888, 6.030918476417832256864518940500, 6.901121058207493895183419426118, 7.51894210741460610687390809344, 8.78635285337447533752379890643, 9.67569197991635069843282042533, 10.22300874076047239503878129562, 10.955488373504562158533856216447, 11.47913788690423762530881600374, 12.43802294316518307376156779919, 13.03438046547863675302049646405, 13.76362257756315974460377849222, 14.54846780603130474340632132591, 14.97069394141009743095176019926, 15.615732703597131609744738547676, 16.70176582535414169897436624985, 17.59690995055975868384552874389, 18.27264196801293520976643871150, 18.883919954383518791934114400993, 19.62011562054472608144249480619

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