↑

Properties

Label 1-449-449.133-r1-0-0
Degree $1$
Conductor $449$
Sign $0.613 - 0.789i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.303 + 0.952i)2-s + (−0.999 + 0.0420i)3-s + (−0.815 − 0.578i)4-s + (−0.623 + 0.781i)5-s + (0.263 − 0.964i)6-s + (−0.276 − 0.960i)7-s + (0.799 − 0.601i)8-s + (0.996 − 0.0840i)9-s + (−0.555 − 0.831i)10-s + (0.111 + 0.993i)11-s + (0.839 + 0.543i)12-s + (−0.125 − 0.992i)13-s + (0.999 + 0.0280i)14-s + (0.590 − 0.807i)15-s + (0.330 + 0.943i)16-s + (0.446 + 0.894i)17-s + ⋯
L(s)  = 1  + (−0.303 + 0.952i)2-s + (−0.999 + 0.0420i)3-s + (−0.815 − 0.578i)4-s + (−0.623 + 0.781i)5-s + (0.263 − 0.964i)6-s + (−0.276 − 0.960i)7-s + (0.799 − 0.601i)8-s + (0.996 − 0.0840i)9-s + (−0.555 − 0.831i)10-s + (0.111 + 0.993i)11-s + (0.839 + 0.543i)12-s + (−0.125 − 0.992i)13-s + (0.999 + 0.0280i)14-s + (0.590 − 0.807i)15-s + (0.330 + 0.943i)16-s + (0.446 + 0.894i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 449 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.613 - 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 449 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.613 - 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(449\)
Sign: $0.613 - 0.789i$
Analytic conductor: \(48.2517\)
Root analytic conductor: \(48.2517\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{449} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 449,\ (1:\ ),\ 0.613 - 0.789i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1133222123 - 0.05547333290i\)
\(L(\frac12)\) \(\approx\) \(0.1133222123 - 0.05547333290i\)
\(L(1)\) \(\approx\) \(0.4109315838 + 0.2456012452i\)
\(L(1)\) \(\approx\) \(0.4109315838 + 0.2456012452i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad449 \( 1 \)
good2 \( 1 + (-0.303 + 0.952i)T \)
3 \( 1 + (-0.999 + 0.0420i)T \)
5 \( 1 + (-0.623 + 0.781i)T \)
7 \( 1 + (-0.276 - 0.960i)T \)
11 \( 1 + (0.111 + 0.993i)T \)
13 \( 1 + (-0.125 - 0.992i)T \)
17 \( 1 + (0.446 + 0.894i)T \)
19 \( 1 + (-0.395 + 0.918i)T \)
23 \( 1 + (-0.167 + 0.985i)T \)
29 \( 1 + (-0.567 - 0.823i)T \)
31 \( 1 + (-0.567 + 0.823i)T \)
37 \( 1 + (0.0980 - 0.995i)T \)
41 \( 1 + (0.912 + 0.408i)T \)
43 \( 1 + (0.543 + 0.839i)T \)
47 \( 1 + (-0.964 - 0.263i)T \)
53 \( 1 + (-0.408 + 0.912i)T \)
59 \( 1 + (-0.726 + 0.686i)T \)
61 \( 1 + (-0.458 + 0.888i)T \)
67 \( 1 + iT \)
71 \( 1 + (0.995 - 0.0980i)T \)
73 \( 1 + (0.929 - 0.369i)T \)
79 \( 1 + (0.881 - 0.471i)T \)
83 \( 1 + (-0.736 - 0.676i)T \)
89 \( 1 + (-0.0560 - 0.998i)T \)
97 \( 1 + (0.139 + 0.990i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.95911745071245868976008328696, −22.800461595440310236405320172872, −22.093417418115401252580523386702, −21.4064689517613797879821415000, −20.60217953289835431127273718672, −19.403757607755797942510346982209, −18.80174337527903463292312094172, −18.177653033545514165156786423953, −16.81191336813230109060899957349, −16.5287363494560047893484363435, −15.54134686446967845506928949076, −14.028664239356663969991364735005, −12.852071939445269155036504560848, −12.34891463375639310471223195723, −11.43229618744614598318158300897, −11.09180360481108792159931620005, −9.551160896024901829399210531141, −9.00570131464475023424383510457, −7.95981931697975631353036990909, −6.6393602798952828144013614056, −5.34163525617295344985154253143, −4.62916525987667955947947062780, −3.50070203936818233458097432961, −2.10906448906599833391059527480, −0.770045461430628805888676465040, 0.06679710819454874596191010071, 1.404905463081717224880202509300, 3.68784224188327204713217711197, 4.379915623867141125885891421722, 5.68300607083145006174429839791, 6.44361560914174631888172317295, 7.53950466536491188821909984982, 7.727011440830832893729896949206, 9.63377965331080473397396693679, 10.33640035519918349619289319424, 10.92871902798259136878653805068, 12.34805232465381700108557932513, 13.08493038756512530496431831273, 14.42513019055344988373277924156, 15.113806312020954365795494932517, 15.93606376262014435853425944989, 16.78832251971086320932202680723, 17.576628604836568288243170739887, 18.1213374862246869548738235925, 19.23446083203178594652391134216, 19.8822616530295327555091470071, 21.37566254233803573335198661601, 22.537974143672391468998534643380, 23.083955664321729708699763169508, 23.3141865824389940873666028110

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