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Properties

Label 1-449-449.179-r1-0-0
Degree $1$
Conductor $449$
Sign $-0.567 + 0.823i$
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  + (0.645 + 0.764i)2-s + (−0.236 + 0.971i)3-s + (−0.167 + 0.985i)4-s + (−0.623 − 0.781i)5-s + (−0.894 + 0.446i)6-s + (−0.483 − 0.875i)7-s + (−0.861 + 0.508i)8-s + (−0.888 − 0.458i)9-s + (0.195 − 0.980i)10-s + (0.993 + 0.111i)11-s + (−0.918 − 0.395i)12-s + (0.754 + 0.655i)13-s + (0.356 − 0.934i)14-s + (0.906 − 0.421i)15-s + (−0.943 − 0.330i)16-s + (0.868 − 0.495i)17-s + ⋯
L(s)  = 1  + (0.645 + 0.764i)2-s + (−0.236 + 0.971i)3-s + (−0.167 + 0.985i)4-s + (−0.623 − 0.781i)5-s + (−0.894 + 0.446i)6-s + (−0.483 − 0.875i)7-s + (−0.861 + 0.508i)8-s + (−0.888 − 0.458i)9-s + (0.195 − 0.980i)10-s + (0.993 + 0.111i)11-s + (−0.918 − 0.395i)12-s + (0.754 + 0.655i)13-s + (0.356 − 0.934i)14-s + (0.906 − 0.421i)15-s + (−0.943 − 0.330i)16-s + (0.868 − 0.495i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 449 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.567 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 449 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.567 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(449\)
Sign: $-0.567 + 0.823i$
Analytic conductor: \(48.2517\)
Root analytic conductor: \(48.2517\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{449} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 449,\ (1:\ ),\ -0.567 + 0.823i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9663540106 + 1.838915853i\)
\(L(\frac12)\) \(\approx\) \(0.9663540106 + 1.838915853i\)
\(L(1)\) \(\approx\) \(0.9924003585 + 0.7492303412i\)
\(L(1)\) \(\approx\) \(0.9924003585 + 0.7492303412i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad449 \( 1 \)
good2 \( 1 + (0.645 + 0.764i)T \)
3 \( 1 + (-0.236 + 0.971i)T \)
5 \( 1 + (-0.623 - 0.781i)T \)
7 \( 1 + (-0.483 - 0.875i)T \)
11 \( 1 + (0.993 + 0.111i)T \)
13 \( 1 + (0.754 + 0.655i)T \)
17 \( 1 + (0.868 - 0.495i)T \)
19 \( 1 + (0.977 - 0.208i)T \)
23 \( 1 + (0.815 + 0.578i)T \)
29 \( 1 + (-0.716 + 0.697i)T \)
31 \( 1 + (-0.716 - 0.697i)T \)
37 \( 1 + (-0.881 - 0.471i)T \)
41 \( 1 + (0.726 + 0.686i)T \)
43 \( 1 + (-0.395 - 0.918i)T \)
47 \( 1 + (0.446 + 0.894i)T \)
53 \( 1 + (-0.686 + 0.726i)T \)
59 \( 1 + (-0.408 - 0.912i)T \)
61 \( 1 + (0.996 - 0.0840i)T \)
67 \( 1 + iT \)
71 \( 1 + (0.471 + 0.881i)T \)
73 \( 1 + (0.839 + 0.543i)T \)
79 \( 1 + (0.634 - 0.773i)T \)
83 \( 1 + (0.854 - 0.520i)T \)
89 \( 1 + (0.666 + 0.745i)T \)
97 \( 1 + (0.249 + 0.968i)T \)
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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

−23.161560372949915606916236960057, −22.63963136052747759064438339795, −22.16720080496190052910823854435, −20.9339050275509585597613347487, −19.81998825738691247628647292538, −19.21266288181962280558997683945, −18.59304334378710909908619476805, −17.95485671270048083410227422687, −16.50381597131654031206767510888, −15.3461044772318691140143905213, −14.591576096353721546843548329554, −13.76353867285338174991472496795, −12.696334809047112252177384537378, −12.09319582979207897570197202733, −11.385658383491335655599281643449, −10.54735134330879870588773278748, −9.26948447769566349946951623388, −8.18027948367149999065552142036, −6.90036075247531167179342312477, −6.12505453454907597683585991600, −5.337664468536954857724733584598, −3.58659785392872687068313027493, −3.0682405245790463698916653729, −1.79795179397078753047045055644, −0.61364607361479808691825269889, 0.91733897959350920287847994464, 3.488276497832574233656992079377, 3.76260467550418889535573963654, 4.79483792770058779756460118998, 5.647997713913240491067176522500, 6.84461917877295959942207680783, 7.719516998428402162050244879448, 9.08195874432331869278591137944, 9.41573072359365891018867239368, 11.10550064340199138691133296539, 11.74609550140119103897312634138, 12.75036175803030773463649261143, 13.82874144585370115817804559120, 14.52654142120384282210136495271, 15.64282906557734635399954778664, 16.268614833523073180761435703485, 16.76421251052238995265800107946, 17.48423220845388010163273738600, 19.02781071126496898825719701080, 20.36192670672924105516762575085, 20.558301300697894905727150855950, 21.71555243070254994531278524080, 22.562459092722750255982150402681, 23.24395225500912559536587822229, 23.80076177065584734838557683276

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