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Properties

Label 1-449-449.157-r1-0-0
Degree $1$
Conductor $449$
Sign $0.166 - 0.986i$
Analytic cond. $48.2517$
Root an. cond. $48.2517$
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  + (0.726 + 0.686i)2-s + (0.567 + 0.823i)3-s + (0.0560 + 0.998i)4-s + (0.222 − 0.974i)5-s + (−0.153 + 0.988i)6-s + (0.167 − 0.985i)7-s + (−0.645 + 0.764i)8-s + (−0.356 + 0.934i)9-s + (0.831 − 0.555i)10-s + (−0.532 − 0.846i)11-s + (−0.790 + 0.612i)12-s + (−0.236 − 0.971i)13-s + (0.799 − 0.601i)14-s + (0.929 − 0.369i)15-s + (−0.993 + 0.111i)16-s + (−0.939 + 0.343i)17-s + ⋯
L(s)  = 1  + (0.726 + 0.686i)2-s + (0.567 + 0.823i)3-s + (0.0560 + 0.998i)4-s + (0.222 − 0.974i)5-s + (−0.153 + 0.988i)6-s + (0.167 − 0.985i)7-s + (−0.645 + 0.764i)8-s + (−0.356 + 0.934i)9-s + (0.831 − 0.555i)10-s + (−0.532 − 0.846i)11-s + (−0.790 + 0.612i)12-s + (−0.236 − 0.971i)13-s + (0.799 − 0.601i)14-s + (0.929 − 0.369i)15-s + (−0.993 + 0.111i)16-s + (−0.939 + 0.343i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 449 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.166 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 449 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.166 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(449\)
Sign: $0.166 - 0.986i$
Analytic conductor: \(48.2517\)
Root analytic conductor: \(48.2517\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{449} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 449,\ (1:\ ),\ 0.166 - 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8529457103 - 0.7211975745i\)
\(L(\frac12)\) \(\approx\) \(0.8529457103 - 0.7211975745i\)
\(L(1)\) \(\approx\) \(1.320609914 + 0.4384150516i\)
\(L(1)\) \(\approx\) \(1.320609914 + 0.4384150516i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad449 \( 1 \)
good2 \( 1 + (0.726 + 0.686i)T \)
3 \( 1 + (0.567 + 0.823i)T \)
5 \( 1 + (0.222 - 0.974i)T \)
7 \( 1 + (0.167 - 0.985i)T \)
11 \( 1 + (-0.532 - 0.846i)T \)
13 \( 1 + (-0.236 - 0.971i)T \)
17 \( 1 + (-0.939 + 0.343i)T \)
19 \( 1 + (-0.0700 + 0.997i)T \)
23 \( 1 + (-0.666 - 0.745i)T \)
29 \( 1 + (-0.964 + 0.263i)T \)
31 \( 1 + (-0.964 - 0.263i)T \)
37 \( 1 + (0.773 - 0.634i)T \)
41 \( 1 + (-0.968 + 0.249i)T \)
43 \( 1 + (0.612 - 0.790i)T \)
47 \( 1 + (0.988 + 0.153i)T \)
53 \( 1 + (-0.249 - 0.968i)T \)
59 \( 1 + (-0.139 + 0.990i)T \)
61 \( 1 + (-0.999 - 0.0280i)T \)
67 \( 1 + iT \)
71 \( 1 + (0.634 - 0.773i)T \)
73 \( 1 + (0.754 - 0.655i)T \)
79 \( 1 + (-0.290 + 0.956i)T \)
83 \( 1 + (-0.181 + 0.983i)T \)
89 \( 1 + (0.960 - 0.276i)T \)
97 \( 1 + (0.0840 - 0.996i)T \)
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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

−23.89830687325840891897783667478, −23.14829600290019770892075147414, −22.024089126800491320828774390214, −21.66149125221105009595501744914, −20.47578969998664585725339126370, −19.752589340224874075233474364367, −18.78646615832056086543254566170, −18.35475150139745002720012468789, −17.5757938489468071385736187830, −15.514161568233111949411533824995, −15.12011600798066531525506369914, −14.204787001821779988810047086563, −13.485084642670743369421708284, −12.62521409672085173676444631815, −11.69293729133942095839740758643, −11.074722201999448101716049275032, −9.63326183953623480473147990337, −9.095182476196409434261841308894, −7.50190285904809327471744855928, −6.69775396524747402505128617755, −5.80818721002185225136337428300, −4.56066393041895175319290595752, −3.23159474506319022557339131633, −2.24415269235009812591827127068, −1.92455430986988469009760217086, 0.17952790924325413474964753147, 2.18375171451334871738937747642, 3.570350472884369712838592795991, 4.21274585799350604928000170604, 5.20743093189362669664832004263, 5.95103853738016757243471806755, 7.59574495980324059833105382703, 8.19912750366955724951958621486, 9.0235969189657899975946473141, 10.282982342167133964934335310530, 11.11349794582153898151830407347, 12.5917758274905134488082548669, 13.29930032158697412550263242593, 13.98622609983768866229023127944, 14.88241373990931524858651012700, 15.80773503049869812912315438301, 16.58020897720663394462816082684, 16.99472256603097820163776428546, 18.18444877254853491139589442527, 19.84880292605999080729039060, 20.426076955443730304300012928572, 20.992863264054937753527377264074, 21.934743073545344550790065805869, 22.64944109266219688356285649855, 23.83711417199660003670977096170

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