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Properties

Label 1-1441-1441.290-r0-0-0
Degree $1$
Conductor $1441$
Sign $-0.734 - 0.678i$
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  + (0.958 − 0.285i)2-s + (0.715 − 0.698i)3-s + (0.836 − 0.548i)4-s + (−0.989 + 0.144i)5-s + (0.485 − 0.873i)6-s + (0.120 − 0.992i)7-s + (0.644 − 0.764i)8-s + (0.0241 − 0.999i)9-s + (−0.906 + 0.421i)10-s + (0.215 − 0.976i)12-s + (0.120 − 0.992i)13-s + (−0.168 − 0.985i)14-s + (−0.607 + 0.794i)15-s + (0.399 − 0.916i)16-s + (0.215 + 0.976i)17-s + (−0.262 − 0.964i)18-s + ⋯
L(s)  = 1  + (0.958 − 0.285i)2-s + (0.715 − 0.698i)3-s + (0.836 − 0.548i)4-s + (−0.989 + 0.144i)5-s + (0.485 − 0.873i)6-s + (0.120 − 0.992i)7-s + (0.644 − 0.764i)8-s + (0.0241 − 0.999i)9-s + (−0.906 + 0.421i)10-s + (0.215 − 0.976i)12-s + (0.120 − 0.992i)13-s + (−0.168 − 0.985i)14-s + (−0.607 + 0.794i)15-s + (0.399 − 0.916i)16-s + (0.215 + 0.976i)17-s + (−0.262 − 0.964i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.734 - 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.734 - 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.734 - 0.678i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (290, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ -0.734 - 0.678i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.174366331 - 2.999975028i\)
\(L(\frac12)\) \(\approx\) \(1.174366331 - 2.999975028i\)
\(L(1)\) \(\approx\) \(1.641172789 - 1.277517113i\)
\(L(1)\) \(\approx\) \(1.641172789 - 1.277517113i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.958 - 0.285i)T \)
3 \( 1 + (0.715 - 0.698i)T \)
5 \( 1 + (-0.989 + 0.144i)T \)
7 \( 1 + (0.120 - 0.992i)T \)
13 \( 1 + (0.120 - 0.992i)T \)
17 \( 1 + (0.215 + 0.976i)T \)
19 \( 1 + (0.995 - 0.0965i)T \)
23 \( 1 + (-0.527 + 0.849i)T \)
29 \( 1 + (0.568 - 0.822i)T \)
31 \( 1 + (0.926 + 0.377i)T \)
37 \( 1 + (-0.354 + 0.935i)T \)
41 \( 1 + (-0.861 - 0.506i)T \)
43 \( 1 + (-0.989 + 0.144i)T \)
47 \( 1 + (0.0241 - 0.999i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (0.215 + 0.976i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + (-0.989 - 0.144i)T \)
71 \( 1 + (-0.0724 + 0.997i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (-0.168 + 0.985i)T \)
83 \( 1 + (0.836 + 0.548i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (-0.681 + 0.732i)T \)
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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

−20.96679697053354864498734176794, −20.50849020332725720854184778678, −19.73630631933899622173076778950, −18.96744773875197804766014350154, −18.15560813938512864900892114566, −16.660748173319725053015697797721, −16.15387640056443316933427931795, −15.69263436016065152098102508072, −14.961134628214457621421522100204, −14.260726003440960660588581098616, −13.712475992887945829507776468031, −12.55592072234825282528844263381, −11.83507207008745048304830688818, −11.40398853439091880083708159200, −10.30461542241689922370826013749, −9.16274134581872489264167368335, −8.5253621439953952005164653550, −7.74277687954839311964593917973, −6.961560527527243184528074820984, −5.83191770650896710498911893450, −4.76758317202244759366843903676, −4.50111220119403228770127412556, −3.31230235250087244432861713768, −2.85841977087320142784948995605, −1.74665953990269694369140704626, 0.80245219984675290963921617463, 1.63290433573791357222864737689, 2.987606418721848737643828775120, 3.47786180614820389281674115890, 4.17154718017870346786898776371, 5.25068762006783996374538300810, 6.39016712850167479926632891742, 7.104549056379902256217430050713, 7.84311857260805541865480961275, 8.355294684470206223569725525107, 9.96263162005280317921381883505, 10.436787956743777446273307588559, 11.6472804435533237407787638146, 11.97864752074091514195369158869, 12.978542828095334292863371596358, 13.58670057113519617448153822865, 14.14350740186074751715409584408, 15.25909154771656795274207385430, 15.332710983766690546393849350046, 16.48520960679670726886587355656, 17.506543236912482855948232967868, 18.40887669457078814588653559609, 19.47125076907468528949975088735, 19.64319354703304091561426673465, 20.390417339746665221053739351117

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