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Properties

Label 1-449-449.30-r1-0-0
Degree $1$
Conductor $449$
Sign $0.631 - 0.775i$
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  + (0.999 + 0.0280i)2-s + (0.208 − 0.977i)3-s + (0.998 + 0.0560i)4-s + (0.222 + 0.974i)5-s + (0.236 − 0.971i)6-s + (−0.985 + 0.167i)7-s + (0.996 + 0.0840i)8-s + (−0.912 − 0.408i)9-s + (0.195 + 0.980i)10-s + (0.532 − 0.846i)11-s + (0.263 − 0.964i)12-s + (0.807 + 0.590i)13-s + (−0.990 + 0.139i)14-s + (0.999 − 0.0140i)15-s + (0.993 + 0.111i)16-s + (0.676 − 0.736i)17-s + ⋯
L(s)  = 1  + (0.999 + 0.0280i)2-s + (0.208 − 0.977i)3-s + (0.998 + 0.0560i)4-s + (0.222 + 0.974i)5-s + (0.236 − 0.971i)6-s + (−0.985 + 0.167i)7-s + (0.996 + 0.0840i)8-s + (−0.912 − 0.408i)9-s + (0.195 + 0.980i)10-s + (0.532 − 0.846i)11-s + (0.263 − 0.964i)12-s + (0.807 + 0.590i)13-s + (−0.990 + 0.139i)14-s + (0.999 − 0.0140i)15-s + (0.993 + 0.111i)16-s + (0.676 − 0.736i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 449 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.631 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 449 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.631 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(449\)
Sign: $0.631 - 0.775i$
Analytic conductor: \(48.2517\)
Root analytic conductor: \(48.2517\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{449} (30, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 449,\ (1:\ ),\ 0.631 - 0.775i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.991859346 - 1.897284702i\)
\(L(\frac12)\) \(\approx\) \(3.991859346 - 1.897284702i\)
\(L(1)\) \(\approx\) \(2.149064369 - 0.5049342039i\)
\(L(1)\) \(\approx\) \(2.149064369 - 0.5049342039i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad449 \( 1 \)
good2 \( 1 + (0.999 + 0.0280i)T \)
3 \( 1 + (0.208 - 0.977i)T \)
5 \( 1 + (0.222 + 0.974i)T \)
7 \( 1 + (-0.985 + 0.167i)T \)
11 \( 1 + (0.532 - 0.846i)T \)
13 \( 1 + (0.807 + 0.590i)T \)
17 \( 1 + (0.676 - 0.736i)T \)
19 \( 1 + (-0.446 - 0.894i)T \)
23 \( 1 + (-0.745 - 0.666i)T \)
29 \( 1 + (0.992 - 0.125i)T \)
31 \( 1 + (0.992 + 0.125i)T \)
37 \( 1 + (0.881 - 0.471i)T \)
41 \( 1 + (0.508 + 0.861i)T \)
43 \( 1 + (0.964 - 0.263i)T \)
47 \( 1 + (0.971 + 0.236i)T \)
53 \( 1 + (0.861 - 0.508i)T \)
59 \( 1 + (-0.799 - 0.601i)T \)
61 \( 1 + (0.686 - 0.726i)T \)
67 \( 1 - iT \)
71 \( 1 + (-0.471 + 0.881i)T \)
73 \( 1 + (-0.948 - 0.317i)T \)
79 \( 1 + (-0.634 - 0.773i)T \)
83 \( 1 + (0.543 + 0.839i)T \)
89 \( 1 + (-0.276 + 0.960i)T \)
97 \( 1 + (-0.645 + 0.764i)T \)
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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

−23.5485986040319832378624740336, −23.017070388560595304522467318032, −22.19322921689797222213873753061, −21.31044442779855248080739093409, −20.66161456132221877914823349963, −19.9643104833734035767068626907, −19.3055657946091538708924800743, −17.414387176418782745377855245707, −16.67717461018354396786397511654, −15.9349949454411680851245826908, −15.32391138940905688988399896047, −14.26607088124768404613167144127, −13.43901023561571754600091627285, −12.53447774556025706943712310281, −11.86710394067456802010226982044, −10.36134542181891592511381696073, −9.98723302309512623533811750508, −8.77204317389033769158658199584, −7.71388008822257903520616396539, −6.107275872215959542626165796125, −5.67424114480277826306105450485, −4.26715667634610738767099972205, −3.91567479346338602706447440123, −2.68356530124299277258953092458, −1.23423806748312744914873099764, 0.88517754626544356254162722681, 2.44771953152131207116846595746, 2.98826434276378045226788016891, 4.027466641706181102364531251551, 5.828094877584363137716753239239, 6.39652954302081303871907688662, 6.921130464112227436052595878600, 8.11835967341214557509675845770, 9.3771404105626547564991794476, 10.7029360297266631283200668209, 11.5622082732601263539057031371, 12.30896113308436935276738546566, 13.42182571295404276295498877111, 13.88407733543816663251733453367, 14.575797887472548885061472910461, 15.75684627241200882406101345105, 16.52584980316015067102491892293, 17.73315872847646500138138150138, 18.886528863566168608843158946369, 19.21909845899048383163260649674, 20.18542695200751700069374391982, 21.38879140361198314896836652014, 22.07225322346130647497873946137, 23.00496236744218497865759734884, 23.40365745028147553645210929924

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