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Properties

Label 1-1441-1441.1003-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.475 + 0.879i$
Analytic cond. $6.69197$
Root an. cond. $6.69197$
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  + (0.926 + 0.377i)2-s + (0.981 + 0.192i)3-s + (0.715 + 0.698i)4-s + (−0.681 − 0.732i)5-s + (0.836 + 0.548i)6-s + (0.998 + 0.0483i)7-s + (0.399 + 0.916i)8-s + (0.926 + 0.377i)9-s + (−0.354 − 0.935i)10-s + (0.568 + 0.822i)12-s + (−0.836 + 0.548i)13-s + (0.906 + 0.421i)14-s + (−0.527 − 0.849i)15-s + (0.0241 + 0.999i)16-s + (0.0241 + 0.999i)17-s + (0.715 + 0.698i)18-s + ⋯
L(s)  = 1  + (0.926 + 0.377i)2-s + (0.981 + 0.192i)3-s + (0.715 + 0.698i)4-s + (−0.681 − 0.732i)5-s + (0.836 + 0.548i)6-s + (0.998 + 0.0483i)7-s + (0.399 + 0.916i)8-s + (0.926 + 0.377i)9-s + (−0.354 − 0.935i)10-s + (0.568 + 0.822i)12-s + (−0.836 + 0.548i)13-s + (0.906 + 0.421i)14-s + (−0.527 − 0.849i)15-s + (0.0241 + 0.999i)16-s + (0.0241 + 0.999i)17-s + (0.715 + 0.698i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.475 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.475 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.475 + 0.879i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1003, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.475 + 0.879i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.552014853 + 2.118394007i\)
\(L(\frac12)\) \(\approx\) \(3.552014853 + 2.118394007i\)
\(L(1)\) \(\approx\) \(2.367657678 + 0.8068908194i\)
\(L(1)\) \(\approx\) \(2.367657678 + 0.8068908194i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.926 + 0.377i)T \)
3 \( 1 + (0.981 + 0.192i)T \)
5 \( 1 + (-0.681 - 0.732i)T \)
7 \( 1 + (0.998 + 0.0483i)T \)
13 \( 1 + (-0.836 + 0.548i)T \)
17 \( 1 + (0.0241 + 0.999i)T \)
19 \( 1 + (0.958 - 0.285i)T \)
23 \( 1 + (0.748 + 0.663i)T \)
29 \( 1 + (-0.0724 - 0.997i)T \)
31 \( 1 + (-0.995 + 0.0965i)T \)
37 \( 1 + (-0.715 - 0.698i)T \)
41 \( 1 + (-0.958 + 0.285i)T \)
43 \( 1 + (-0.120 - 0.992i)T \)
47 \( 1 + (0.0724 - 0.997i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (0.958 + 0.285i)T \)
61 \( 1 + (0.809 - 0.587i)T \)
67 \( 1 + (-0.120 + 0.992i)T \)
71 \( 1 + (-0.215 + 0.976i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (0.485 - 0.873i)T \)
83 \( 1 + (-0.989 + 0.144i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.836 + 0.548i)T \)
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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

−20.48382735487458443651941027653, −20.10562684919362109807283661121, −19.2873010118660994011357691838, −18.507658774053616690981889132833, −18.010616427306251484558021376017, −16.55070870287509705530596862712, −15.61135543299743772010464283229, −15.04889184065045778352200501778, −14.32017938214425187321001531459, −14.11736123651905699258739594654, −12.99831845314933442784889651828, −12.21586418575095480832079728653, −11.54208446208451206475876526921, −10.721403193110303895834579438700, −9.977785865600484795359767433504, −8.959821010998458062979896611110, −7.75414030907246673134094377003, −7.39749656114972974187094147721, −6.607316860751225142823989949115, −5.15272340606173054082117239869, −4.6555960656423849250539653379, −3.48870828005273074515368592178, −2.99789648585842613179900392745, −2.14808293494689536899140814560, −1.07619357502200702744331695657, 1.52310872773547386337221353919, 2.25326244029816944626852545093, 3.51520906034276937470254941181, 4.03198337087783235640940302592, 4.951964758338092755329205670716, 5.40612745858216859149093451657, 7.08069313311315776049295978652, 7.4675065133628081633721595262, 8.3517313389652368645976722418, 8.84924947906072824574344800237, 9.98927250867328116960076362739, 11.205470620207083351913100760744, 11.78692563640684849894434283010, 12.63678348146314802092685466464, 13.34387419717376060020903148006, 14.13299118843963692271838524607, 14.83497026016393192217706906288, 15.32403193918646416592653474957, 16.05285299411437149443987535240, 16.92372108831539201888506811360, 17.55615731208434103707294726668, 18.91123254153988185963494200117, 19.60009120311050387904649601541, 20.336267042530387849608412833362, 20.82022183513394870193665930179

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