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Properties

Label 1-1441-1441.758-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.260 + 0.965i$
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  + (0.0241 − 0.999i)2-s + (−0.168 + 0.985i)3-s + (−0.998 − 0.0483i)4-s + (−0.443 − 0.896i)5-s + (0.981 + 0.192i)6-s + (0.681 − 0.732i)7-s + (−0.0724 + 0.997i)8-s + (−0.943 − 0.331i)9-s + (−0.906 + 0.421i)10-s + (0.215 − 0.976i)12-s + (0.906 + 0.421i)13-s + (−0.715 − 0.698i)14-s + (0.958 − 0.285i)15-s + (0.995 + 0.0965i)16-s + (−0.861 + 0.506i)17-s + (−0.354 + 0.935i)18-s + ⋯
L(s)  = 1  + (0.0241 − 0.999i)2-s + (−0.168 + 0.985i)3-s + (−0.998 − 0.0483i)4-s + (−0.443 − 0.896i)5-s + (0.981 + 0.192i)6-s + (0.681 − 0.732i)7-s + (−0.0724 + 0.997i)8-s + (−0.943 − 0.331i)9-s + (−0.906 + 0.421i)10-s + (0.215 − 0.976i)12-s + (0.906 + 0.421i)13-s + (−0.715 − 0.698i)14-s + (0.958 − 0.285i)15-s + (0.995 + 0.0965i)16-s + (−0.861 + 0.506i)17-s + (−0.354 + 0.935i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.260 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.260 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.260 + 0.965i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (758, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.260 + 0.965i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3449359934 + 0.2641952208i\)
\(L(\frac12)\) \(\approx\) \(0.3449359934 + 0.2641952208i\)
\(L(1)\) \(\approx\) \(0.7004852362 - 0.2023570654i\)
\(L(1)\) \(\approx\) \(0.7004852362 - 0.2023570654i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.0241 - 0.999i)T \)
3 \( 1 + (-0.168 + 0.985i)T \)
5 \( 1 + (-0.443 - 0.896i)T \)
7 \( 1 + (0.681 - 0.732i)T \)
13 \( 1 + (0.906 + 0.421i)T \)
17 \( 1 + (-0.861 + 0.506i)T \)
19 \( 1 + (-0.748 + 0.663i)T \)
23 \( 1 + (0.527 - 0.849i)T \)
29 \( 1 + (-0.943 + 0.331i)T \)
31 \( 1 + (-0.644 + 0.764i)T \)
37 \( 1 + (-0.836 + 0.548i)T \)
41 \( 1 + (-0.995 + 0.0965i)T \)
43 \( 1 + (0.989 - 0.144i)T \)
47 \( 1 + (-0.568 - 0.822i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (0.399 - 0.916i)T \)
61 \( 1 + (-0.309 + 0.951i)T \)
67 \( 1 + (0.989 + 0.144i)T \)
71 \( 1 + (0.970 - 0.239i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (0.885 + 0.464i)T \)
83 \( 1 + (-0.262 + 0.964i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (-0.485 - 0.873i)T \)
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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

−20.518928091036006985177188729754, −19.363642063338761413778853355469, −18.84385249134304391174815175196, −18.17053033269234667343838402077, −17.71953541145834904183770112534, −17.01330990499537572370730466312, −15.75693087394296981912164308078, −15.32174587705419153074872063806, −14.59387502561101176206078666893, −13.78211506383179948752343615011, −13.150634536418280061531912581, −12.30998003494127321919318945249, −11.1923343280092559225135785479, −10.97960109529483571408535412311, −9.34200824361589772036752588551, −8.63441438193607078779835032470, −7.85954798323561985886099817501, −7.26848236548078480825714479355, −6.457534466977387984543307982279, −5.814366727341139159568231069228, −4.9647882169730854115196606994, −3.799607647056247423922722165682, −2.75533752828806586202375755060, −1.69761993407682291912446452767, −0.183874772100975908002177815334, 1.15126504146569035649111949863, 2.06264749132117128385356402210, 3.72881830781990701122868774517, 3.84140644346392456761551327524, 4.80089717355887069734406285180, 5.33208522585174150077017881062, 6.64052821284680285934029311982, 8.24324619783935236582912253664, 8.5378657574411608331260551007, 9.308046067783699740558184378168, 10.330554619423052366461246788137, 10.967674865365692974698784401130, 11.39121620096038175975349684373, 12.391495612791630907862009409806, 13.10870009281427876126565755094, 14.00569606249161759486012037865, 14.733609896291753842237278667313, 15.580668828411158033256879386, 16.66052296131582102622306178109, 16.96519877676716149841453004348, 17.83169547697556276460348862213, 18.78702737597922866877519396909, 19.73012806063429584507739344099, 20.32304227073659857049106908870, 20.89257662958585474596176923549

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