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Properties

Label 1-449-449.177-r1-0-0
Degree $1$
Conductor $449$
Sign $-0.0970 + 0.995i$
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  + (0.139 − 0.990i)2-s + (−0.964 − 0.263i)3-s + (−0.960 − 0.276i)4-s + (0.900 + 0.433i)5-s + (−0.395 + 0.918i)6-s + (0.666 − 0.745i)7-s + (−0.408 + 0.912i)8-s + (0.861 + 0.508i)9-s + (0.555 − 0.831i)10-s + (0.330 + 0.943i)11-s + (0.854 + 0.520i)12-s + (−0.716 + 0.697i)13-s + (−0.645 − 0.764i)14-s + (−0.754 − 0.655i)15-s + (0.846 + 0.532i)16-s + (0.208 + 0.977i)17-s + ⋯
L(s)  = 1  + (0.139 − 0.990i)2-s + (−0.964 − 0.263i)3-s + (−0.960 − 0.276i)4-s + (0.900 + 0.433i)5-s + (−0.395 + 0.918i)6-s + (0.666 − 0.745i)7-s + (−0.408 + 0.912i)8-s + (0.861 + 0.508i)9-s + (0.555 − 0.831i)10-s + (0.330 + 0.943i)11-s + (0.854 + 0.520i)12-s + (−0.716 + 0.697i)13-s + (−0.645 − 0.764i)14-s + (−0.754 − 0.655i)15-s + (0.846 + 0.532i)16-s + (0.208 + 0.977i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 449 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0970 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 449 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0970 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(449\)
Sign: $-0.0970 + 0.995i$
Analytic conductor: \(48.2517\)
Root analytic conductor: \(48.2517\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{449} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 449,\ (1:\ ),\ -0.0970 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2190094421 + 0.2413920356i\)
\(L(\frac12)\) \(\approx\) \(0.2190094421 + 0.2413920356i\)
\(L(1)\) \(\approx\) \(0.7279591195 - 0.3070787702i\)
\(L(1)\) \(\approx\) \(0.7279591195 - 0.3070787702i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad449 \( 1 \)
good2 \( 1 + (0.139 - 0.990i)T \)
3 \( 1 + (-0.964 - 0.263i)T \)
5 \( 1 + (0.900 + 0.433i)T \)
7 \( 1 + (0.666 - 0.745i)T \)
11 \( 1 + (0.330 + 0.943i)T \)
13 \( 1 + (-0.716 + 0.697i)T \)
17 \( 1 + (0.208 + 0.977i)T \)
19 \( 1 + (0.999 + 0.0420i)T \)
23 \( 1 + (-0.875 - 0.483i)T \)
29 \( 1 + (-0.988 - 0.153i)T \)
31 \( 1 + (-0.988 + 0.153i)T \)
37 \( 1 + (-0.995 + 0.0980i)T \)
41 \( 1 + (-0.888 - 0.458i)T \)
43 \( 1 + (-0.520 - 0.854i)T \)
47 \( 1 + (-0.918 - 0.395i)T \)
53 \( 1 + (-0.458 + 0.888i)T \)
59 \( 1 + (-0.0840 + 0.996i)T \)
61 \( 1 + (-0.799 - 0.601i)T \)
67 \( 1 - iT \)
71 \( 1 + (0.0980 - 0.995i)T \)
73 \( 1 + (-0.736 + 0.676i)T \)
79 \( 1 + (0.471 - 0.881i)T \)
83 \( 1 + (-0.868 + 0.495i)T \)
89 \( 1 + (0.985 - 0.167i)T \)
97 \( 1 + (-0.934 - 0.356i)T \)
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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

−23.832135435463790674368892273806, −22.51854197230115465403972168632, −22.05629661505199643110106427280, −21.44508690684386812642539630130, −20.42904335901741562587214220118, −18.72929763368071281388175458221, −17.984043621395109817529230737664, −17.52468676857714520445577188391, −16.519324119246254547970245673, −16.03401886268366415664678476606, −14.92269859709453630163353739181, −14.08155295290254220597570682246, −13.12097729929053752604382997715, −12.17588808049949363709486294941, −11.333015202550922001483105884041, −9.873052982921476921267223000530, −9.33873677564495913459406132284, −8.209847620378834894239684190121, −7.110114953760186824243210020875, −5.89963856369144676286252711226, −5.41984304551975432846265531649, −4.82779465522952265971825275070, −3.29325936439608109609700549147, −1.42806866383714330108973933941, −0.093043166069770910686034347469, 1.624408002360941688464412617934, 1.87010083281124675674794892476, 3.7217489779489409303713828004, 4.74677589590608341767864419352, 5.52900549775497115675219873012, 6.726859937814645859520030496577, 7.66517290272909512871270344061, 9.28522657215482229460587514637, 10.17611357040457656526971267071, 10.64601848983597522655508939596, 11.72084323385170874204324814781, 12.36365606263428222973094443051, 13.409155605175190426881463490, 14.165856469970237949136675621410, 14.97173222897480258525679970908, 16.82168115713644320215351300171, 17.24762401324620098631919037276, 18.06232495219794279589231678246, 18.66752968546294472307128591842, 19.84755348910038640841951756970, 20.67512324525997872211782415190, 21.62869374912830977536252729938, 22.19781219975093792115724042925, 22.899907954127410524634485967115, 23.885788173491866830657693740788

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