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Properties

Label 1-1441-1441.1084-r1-0-0
Degree $1$
Conductor $1441$
Sign $-0.445 + 0.895i$
Analytic cond. $154.856$
Root an. cond. $154.856$
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  + (0.989 + 0.144i)2-s + (0.926 + 0.377i)3-s + (0.958 + 0.285i)4-s + (−0.0724 + 0.997i)5-s + (0.861 + 0.506i)6-s + (0.748 + 0.663i)7-s + (0.906 + 0.421i)8-s + (0.715 + 0.698i)9-s + (−0.215 + 0.976i)10-s + (0.779 + 0.626i)12-s + (0.748 + 0.663i)13-s + (0.644 + 0.764i)14-s + (−0.443 + 0.896i)15-s + (0.836 + 0.548i)16-s + (−0.779 + 0.626i)17-s + (0.607 + 0.794i)18-s + ⋯
L(s)  = 1  + (0.989 + 0.144i)2-s + (0.926 + 0.377i)3-s + (0.958 + 0.285i)4-s + (−0.0724 + 0.997i)5-s + (0.861 + 0.506i)6-s + (0.748 + 0.663i)7-s + (0.906 + 0.421i)8-s + (0.715 + 0.698i)9-s + (−0.215 + 0.976i)10-s + (0.779 + 0.626i)12-s + (0.748 + 0.663i)13-s + (0.644 + 0.764i)14-s + (−0.443 + 0.896i)15-s + (0.836 + 0.548i)16-s + (−0.779 + 0.626i)17-s + (0.607 + 0.794i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.445 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.445 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.445 + 0.895i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1084, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ -0.445 + 0.895i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.444939750 + 7.181101891i\)
\(L(\frac12)\) \(\approx\) \(4.444939750 + 7.181101891i\)
\(L(1)\) \(\approx\) \(2.713971982 + 1.754674273i\)
\(L(1)\) \(\approx\) \(2.713971982 + 1.754674273i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.989 + 0.144i)T \)
3 \( 1 + (0.926 + 0.377i)T \)
5 \( 1 + (-0.0724 + 0.997i)T \)
7 \( 1 + (0.748 + 0.663i)T \)
13 \( 1 + (0.748 + 0.663i)T \)
17 \( 1 + (-0.779 + 0.626i)T \)
19 \( 1 + (0.998 + 0.0483i)T \)
23 \( 1 + (0.485 - 0.873i)T \)
29 \( 1 + (-0.885 - 0.464i)T \)
31 \( 1 + (0.981 - 0.192i)T \)
37 \( 1 + (0.568 - 0.822i)T \)
41 \( 1 + (0.262 + 0.964i)T \)
43 \( 1 + (0.0724 - 0.997i)T \)
47 \( 1 + (0.715 + 0.698i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (0.779 - 0.626i)T \)
61 \( 1 + (-0.309 - 0.951i)T \)
67 \( 1 + (-0.0724 - 0.997i)T \)
71 \( 1 + (-0.681 + 0.732i)T \)
73 \( 1 + (-0.309 - 0.951i)T \)
79 \( 1 + (-0.644 + 0.764i)T \)
83 \( 1 + (-0.958 + 0.285i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (0.399 - 0.916i)T \)
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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

−20.35235212640837897182054676589, −20.02775336013248678921496611076, −19.09865475052100182574846508778, −18.03862889604490223620447504009, −17.291113168314592710367294359748, −16.18062799065473340227388955377, −15.61878385363693478062531016077, −14.87794722497972116804734790786, −13.94793214240001068115729932290, −13.38697569803736942238873093583, −13.067812327522445866174991234206, −11.94975501452735113952371289566, −11.38783903837099487530784579100, −10.34472118083176809088241166965, −9.36592344597179200043195250639, −8.49613581406154060569359197313, −7.63177679709473922310780965490, −7.13744288020236900591335171691, −5.88910457812030943742853793117, −5.00946010856472576531493846421, −4.25956667803084443405740206758, −3.489883412301197349482936698168, −2.55349341662576821633400339942, −1.34134539222756303384877199785, −1.0047723327883188772169554288, 1.64256755123473140452221051446, 2.3582284846751205280981788356, 3.09479404558320889224767224533, 4.00683357275313227644275202824, 4.63130779347062723986029206529, 5.77015018610717932841125881432, 6.55759824414441629534742706502, 7.47839723463584985822168604919, 8.16527335663069281127094142023, 9.034373817932793248431443133312, 10.10558376479294513167514732007, 11.15283074749021270775658366247, 11.33282931487598689493807041028, 12.56258816617342682505491522169, 13.4246271963328887100542577075, 14.199304507225305829369246106478, 14.53553195444689594584678697730, 15.46116258772644117982765879821, 15.661604910348788149974416824166, 16.77104511953333362124472392115, 17.88727391949042407422145109496, 18.74123917672202638219237543621, 19.30840783538312354924212368333, 20.34681243226239066345702923273, 20.884511963454179919039956380922

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