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Properties

Label 1-1441-1441.714-r1-0-0
Degree $1$
Conductor $1441$
Sign $0.986 - 0.164i$
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  + (0.681 + 0.732i)2-s + (−0.861 + 0.506i)3-s + (−0.0724 + 0.997i)4-s + (0.995 + 0.0965i)5-s + (−0.958 − 0.285i)6-s + (0.943 + 0.331i)7-s + (−0.779 + 0.626i)8-s + (0.485 − 0.873i)9-s + (0.607 + 0.794i)10-s + (−0.443 − 0.896i)12-s + (0.607 − 0.794i)13-s + (0.399 + 0.916i)14-s + (−0.906 + 0.421i)15-s + (−0.989 − 0.144i)16-s + (−0.715 − 0.698i)17-s + (0.970 − 0.239i)18-s + ⋯
L(s)  = 1  + (0.681 + 0.732i)2-s + (−0.861 + 0.506i)3-s + (−0.0724 + 0.997i)4-s + (0.995 + 0.0965i)5-s + (−0.958 − 0.285i)6-s + (0.943 + 0.331i)7-s + (−0.779 + 0.626i)8-s + (0.485 − 0.873i)9-s + (0.607 + 0.794i)10-s + (−0.443 − 0.896i)12-s + (0.607 − 0.794i)13-s + (0.399 + 0.916i)14-s + (−0.906 + 0.421i)15-s + (−0.989 − 0.144i)16-s + (−0.715 − 0.698i)17-s + (0.970 − 0.239i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.986 - 0.164i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (714, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ 0.986 - 0.164i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.923708279 - 0.1594490190i\)
\(L(\frac12)\) \(\approx\) \(1.923708279 - 0.1594490190i\)
\(L(1)\) \(\approx\) \(1.165153869 + 0.6148643182i\)
\(L(1)\) \(\approx\) \(1.165153869 + 0.6148643182i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.681 + 0.732i)T \)
3 \( 1 + (-0.861 + 0.506i)T \)
5 \( 1 + (0.995 + 0.0965i)T \)
7 \( 1 + (0.943 + 0.331i)T \)
13 \( 1 + (0.607 - 0.794i)T \)
17 \( 1 + (-0.715 - 0.698i)T \)
19 \( 1 + (-0.885 - 0.464i)T \)
23 \( 1 + (-0.998 - 0.0483i)T \)
29 \( 1 + (-0.485 - 0.873i)T \)
31 \( 1 + (-0.262 + 0.964i)T \)
37 \( 1 + (0.644 - 0.764i)T \)
41 \( 1 + (0.989 - 0.144i)T \)
43 \( 1 + (-0.215 - 0.976i)T \)
47 \( 1 + (0.120 + 0.992i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (-0.168 - 0.985i)T \)
61 \( 1 + (-0.309 - 0.951i)T \)
67 \( 1 + (0.215 - 0.976i)T \)
71 \( 1 + (-0.354 - 0.935i)T \)
73 \( 1 + (-0.309 + 0.951i)T \)
79 \( 1 + (0.748 + 0.663i)T \)
83 \( 1 + (-0.926 + 0.377i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (0.0241 - 0.999i)T \)
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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

−20.75412141393394248598775675404, −19.957027425082587610705917784790, −18.966843374400678546277055764686, −18.24841692115155103390883491174, −17.781501971683739384074722927383, −16.89812779930511198300026076755, −16.20104798508374608361457812247, −14.87771203233225426168887366812, −14.33234309882652673372751434942, −13.33512579466633182710937853028, −13.14296685856517290090385845550, −12.09531383892355368396068631102, −11.34928891027624083583006787684, −10.75853778021056495834012857225, −10.14116363444208649408018125899, −9.089810706277574876694425079847, −8.095492557504564515767119379190, −6.81196628069567163524918561645, −6.136090447061684839309695662303, −5.57476388941650468711988825998, −4.54431925682982668959245330482, −4.0528452617497006735308191691, −2.352894804163087253803731575536, −1.73057602884276623425005762326, −1.15224247826071387157991874614, 0.30331436118776940068666081613, 1.82496319719167962324884659920, 2.83027954455963120876964861679, 4.10132680970120326463246610245, 4.76989509976433626934694207601, 5.58395623261891260198921255724, 6.04161582447541943384057441128, 6.8606485495058031173525716815, 7.924540589626617809973742097415, 8.86377807553831977472691198902, 9.581334920765959589326881166374, 10.87754718619989092652820811308, 11.12706055266511624807857734413, 12.30342492681199461648978799665, 12.87372392689829546956374280462, 13.84633075868364170216884488737, 14.44227358728810592872007133102, 15.41478126739049499257561536743, 15.769422001929622019893393142539, 16.81447635291459616294455895474, 17.481432334897819338382978896244, 17.90694767263779787574298411365, 18.4607038662627021369454588027, 20.23223556312092122765128655649, 20.89421016875583915833316350958

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