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Properties

Label 1-1449-1449.121-r0-0-0
Degree $1$
Conductor $1449$
Sign $0.997 + 0.0725i$
Analytic cond. $6.72912$
Root an. cond. $6.72912$
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  + (−0.654 − 0.755i)2-s + (−0.142 + 0.989i)4-s + (0.580 + 0.814i)5-s + (0.841 − 0.540i)8-s + (0.235 − 0.971i)10-s + (0.981 + 0.189i)11-s + (0.723 + 0.690i)13-s + (−0.959 − 0.281i)16-s + (0.928 + 0.371i)17-s + (0.928 − 0.371i)19-s + (−0.888 + 0.458i)20-s + (−0.5 − 0.866i)22-s + (−0.327 + 0.945i)25-s + (0.0475 − 0.998i)26-s + (0.928 + 0.371i)29-s + ⋯
L(s)  = 1  + (−0.654 − 0.755i)2-s + (−0.142 + 0.989i)4-s + (0.580 + 0.814i)5-s + (0.841 − 0.540i)8-s + (0.235 − 0.971i)10-s + (0.981 + 0.189i)11-s + (0.723 + 0.690i)13-s + (−0.959 − 0.281i)16-s + (0.928 + 0.371i)17-s + (0.928 − 0.371i)19-s + (−0.888 + 0.458i)20-s + (−0.5 − 0.866i)22-s + (−0.327 + 0.945i)25-s + (0.0475 − 0.998i)26-s + (0.928 + 0.371i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0725i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0725i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1449\)    =    \(3^{2} \cdot 7 \cdot 23\)
Sign: $0.997 + 0.0725i$
Analytic conductor: \(6.72912\)
Root analytic conductor: \(6.72912\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1449} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1449,\ (0:\ ),\ 0.997 + 0.0725i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.539773494 + 0.05589363393i\)
\(L(\frac12)\) \(\approx\) \(1.539773494 + 0.05589363393i\)
\(L(1)\) \(\approx\) \(1.028009946 - 0.07946568721i\)
\(L(1)\) \(\approx\) \(1.028009946 - 0.07946568721i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.654 - 0.755i)T \)
5 \( 1 + (0.580 + 0.814i)T \)
11 \( 1 + (0.981 + 0.189i)T \)
13 \( 1 + (0.723 + 0.690i)T \)
17 \( 1 + (0.928 + 0.371i)T \)
19 \( 1 + (0.928 - 0.371i)T \)
29 \( 1 + (0.928 + 0.371i)T \)
31 \( 1 + (0.841 - 0.540i)T \)
37 \( 1 + (0.580 - 0.814i)T \)
41 \( 1 + (-0.995 + 0.0950i)T \)
43 \( 1 + (-0.888 + 0.458i)T \)
47 \( 1 + T \)
53 \( 1 + (0.723 - 0.690i)T \)
59 \( 1 + (-0.959 + 0.281i)T \)
61 \( 1 + (0.841 - 0.540i)T \)
67 \( 1 + (-0.654 - 0.755i)T \)
71 \( 1 + (-0.654 - 0.755i)T \)
73 \( 1 + (-0.786 - 0.618i)T \)
79 \( 1 + (-0.959 + 0.281i)T \)
83 \( 1 + (-0.995 - 0.0950i)T \)
89 \( 1 + (0.0475 - 0.998i)T \)
97 \( 1 + (0.580 + 0.814i)T \)
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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

−20.35248919609318212902040999579, −20.10641663499428305998566123673, −18.98277578169934388910062454117, −18.360810835622770478993527471052, −17.5029024495672163485562054302, −17.0053532879463527519384879735, −16.249657670065013388059715218615, −15.68886923675727275187520928852, −14.69236842999941428971806951964, −13.84907508769293474547014292957, −13.471112579669202262072628583540, −12.19337328950780756766337645464, −11.554244081390797430625330846874, −10.16558563333051143484004799221, −9.95575889688656840637950249293, −8.841877884447511455195351041127, −8.44599596542954835641706446777, −7.5100690498150430949073546658, −6.49823766547497550878512666749, −5.788908411591514046317262705270, −5.149550190176692837985860248455, −4.14049134683054818104804460836, −2.87223082482831218025884320469, −1.334588790723207289160575740637, −1.01919628408633615323635488857, 1.12661664851403672285679639513, 1.83751621334207387897139642074, 2.944388561978674871196012266594, 3.59548157429162785978934804524, 4.57203509766456436536903390619, 5.92129069820764272484030219442, 6.73209871906303816513598564149, 7.454744443845166355807083408325, 8.49833723158225197839440819859, 9.29542323823603521254339002269, 9.93909654154070253140537942781, 10.62003187456884673513407444918, 11.58443701488875831719580599251, 11.92188451162440880561593850830, 13.09530567683214314745873656035, 13.85427651534784862615298880181, 14.426752533809944699009138448577, 15.51674447312881227997963322938, 16.514725603887284567214322258297, 17.096500124770219404711761978196, 17.93102647627126632636737053872, 18.46123991933023525364491340648, 19.18611447671022068082445718758, 19.84199758860744157388608458412, 20.73192423812350991427326721428

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