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Properties

Label 1-1449-1449.1129-r0-0-0
Degree $1$
Conductor $1449$
Sign $0.0547 + 0.998i$
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  + (0.415 + 0.909i)2-s + (−0.654 + 0.755i)4-s + (−0.888 + 0.458i)5-s + (−0.959 − 0.281i)8-s + (−0.786 − 0.618i)10-s + (−0.995 + 0.0950i)11-s + (0.928 − 0.371i)13-s + (−0.142 − 0.989i)16-s + (0.981 − 0.189i)17-s + (0.981 + 0.189i)19-s + (0.235 − 0.971i)20-s + (−0.5 − 0.866i)22-s + (0.580 − 0.814i)25-s + (0.723 + 0.690i)26-s + (0.981 − 0.189i)29-s + ⋯
L(s)  = 1  + (0.415 + 0.909i)2-s + (−0.654 + 0.755i)4-s + (−0.888 + 0.458i)5-s + (−0.959 − 0.281i)8-s + (−0.786 − 0.618i)10-s + (−0.995 + 0.0950i)11-s + (0.928 − 0.371i)13-s + (−0.142 − 0.989i)16-s + (0.981 − 0.189i)17-s + (0.981 + 0.189i)19-s + (0.235 − 0.971i)20-s + (−0.5 − 0.866i)22-s + (0.580 − 0.814i)25-s + (0.723 + 0.690i)26-s + (0.981 − 0.189i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0547 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0547 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1449\)    =    \(3^{2} \cdot 7 \cdot 23\)
Sign: $0.0547 + 0.998i$
Analytic conductor: \(6.72912\)
Root analytic conductor: \(6.72912\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1449} (1129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1449,\ (0:\ ),\ 0.0547 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.019221840 + 0.9648381627i\)
\(L(\frac12)\) \(\approx\) \(1.019221840 + 0.9648381627i\)
\(L(1)\) \(\approx\) \(0.8994821362 + 0.5618623162i\)
\(L(1)\) \(\approx\) \(0.8994821362 + 0.5618623162i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.415 + 0.909i)T \)
5 \( 1 + (-0.888 + 0.458i)T \)
11 \( 1 + (-0.995 + 0.0950i)T \)
13 \( 1 + (0.928 - 0.371i)T \)
17 \( 1 + (0.981 - 0.189i)T \)
19 \( 1 + (0.981 + 0.189i)T \)
29 \( 1 + (0.981 - 0.189i)T \)
31 \( 1 + (-0.959 - 0.281i)T \)
37 \( 1 + (-0.888 - 0.458i)T \)
41 \( 1 + (0.0475 - 0.998i)T \)
43 \( 1 + (0.235 - 0.971i)T \)
47 \( 1 + T \)
53 \( 1 + (0.928 + 0.371i)T \)
59 \( 1 + (-0.142 + 0.989i)T \)
61 \( 1 + (-0.959 - 0.281i)T \)
67 \( 1 + (0.415 + 0.909i)T \)
71 \( 1 + (0.415 + 0.909i)T \)
73 \( 1 + (-0.327 - 0.945i)T \)
79 \( 1 + (-0.142 + 0.989i)T \)
83 \( 1 + (0.0475 + 0.998i)T \)
89 \( 1 + (0.723 + 0.690i)T \)
97 \( 1 + (-0.888 + 0.458i)T \)
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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

−20.530112391526593617604660698079, −19.93894255977416851713083263020, −19.10563298373207524314691504082, −18.503208218885932305474997598829, −17.86991160404549050364392954872, −16.58199601979755938775249612788, −15.91421823948673460552781047832, −15.246424487635252438749757365224, −14.2617429718999622866490866735, −13.55611275875536399448037011239, −12.76023925798322615068232016343, −12.131306615910879001422413059957, −11.38664118506727339250596712890, −10.72192981255064574218727403531, −9.88515620793453441878894786852, −8.9255647671658183014792047160, −8.23358448207495521217622212946, −7.363385847182634100243661150065, −6.0784707708055306210843543500, −5.20817389867421647435668032837, −4.55916158661941552140524339363, −3.48824589866863216092127077438, −3.04947052692245684983062060329, −1.646480408083637236242251952974, −0.76428750097226327314188246919, 0.74898485764160845783713217313, 2.63705110676040135945868668018, 3.45704213674241718682822801317, 4.08629582460103185539236014715, 5.3035120723281967316991773604, 5.74422379884715100205203692179, 6.987483103083608951029380069420, 7.51917960096788137892896387141, 8.18337793661279457658548079719, 8.96982293740980057822761187269, 10.166336277641477374100061407470, 10.90089575372879901119105170023, 12.02669536502354272775496834869, 12.44301512699563275882231046329, 13.56328300881596796114797249307, 14.066206619338398649678324322811, 15.0010272341428332198401814699, 15.734182688787336824462923886450, 16.008799052699496842445213200436, 16.93608359457123576782151019201, 18.03715566736451245234538029895, 18.411771787391623357721421709603, 19.15149176590879639498955286823, 20.36843760306197742936711878263, 20.89826653832623979065669328873

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