Properties

Label 1-1441-1441.675-r0-0-0
Degree $1$
Conductor $1441$
Sign $-0.684 - 0.729i$
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  + (−0.906 − 0.421i)2-s + (0.399 + 0.916i)3-s + (0.644 + 0.764i)4-s + (0.215 − 0.976i)5-s + (0.0241 − 0.999i)6-s + (0.568 − 0.822i)7-s + (−0.262 − 0.964i)8-s + (−0.681 + 0.732i)9-s + (−0.607 + 0.794i)10-s + (−0.443 + 0.896i)12-s + (0.568 − 0.822i)13-s + (−0.861 + 0.506i)14-s + (0.981 − 0.192i)15-s + (−0.168 + 0.985i)16-s + (−0.443 − 0.896i)17-s + (0.926 − 0.377i)18-s + ⋯
L(s)  = 1  + (−0.906 − 0.421i)2-s + (0.399 + 0.916i)3-s + (0.644 + 0.764i)4-s + (0.215 − 0.976i)5-s + (0.0241 − 0.999i)6-s + (0.568 − 0.822i)7-s + (−0.262 − 0.964i)8-s + (−0.681 + 0.732i)9-s + (−0.607 + 0.794i)10-s + (−0.443 + 0.896i)12-s + (0.568 − 0.822i)13-s + (−0.861 + 0.506i)14-s + (0.981 − 0.192i)15-s + (−0.168 + 0.985i)16-s + (−0.443 − 0.896i)17-s + (0.926 − 0.377i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.684 - 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.684 - 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.684 - 0.729i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (675, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ -0.684 - 0.729i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3008094415 - 0.6944884890i\)
\(L(\frac12)\) \(\approx\) \(0.3008094415 - 0.6944884890i\)
\(L(1)\) \(\approx\) \(0.7290283892 - 0.2134535033i\)
\(L(1)\) \(\approx\) \(0.7290283892 - 0.2134535033i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.906 + 0.421i)T \)
3 \( 1 + (-0.399 - 0.916i)T \)
5 \( 1 + (-0.215 + 0.976i)T \)
7 \( 1 + (-0.568 + 0.822i)T \)
13 \( 1 + (-0.568 + 0.822i)T \)
17 \( 1 + (0.443 + 0.896i)T \)
19 \( 1 + (0.989 + 0.144i)T \)
23 \( 1 + (0.998 - 0.0483i)T \)
29 \( 1 + (-0.120 - 0.992i)T \)
31 \( 1 + (-0.836 + 0.548i)T \)
37 \( 1 + (0.970 + 0.239i)T \)
41 \( 1 + (-0.715 - 0.698i)T \)
43 \( 1 + (-0.215 + 0.976i)T \)
47 \( 1 + (0.681 - 0.732i)T \)
53 \( 1 + (-0.309 + 0.951i)T \)
59 \( 1 + (0.443 + 0.896i)T \)
61 \( 1 + (0.809 + 0.587i)T \)
67 \( 1 + (-0.215 - 0.976i)T \)
71 \( 1 + (-0.779 - 0.626i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (0.861 + 0.506i)T \)
83 \( 1 + (-0.644 + 0.764i)T \)
89 \( 1 + (0.809 - 0.587i)T \)
97 \( 1 + (0.943 - 0.331i)T \)
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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

−21.07414223827602276663296863968, −19.81789926913842527727067445717, −19.173999081110433997653617269088, −18.77704884357918821066162412205, −17.957371539586761584275660567142, −17.645263704243286300505805145144, −16.71967934091952976917475211675, −15.401645960178000474312991497594, −15.15679828545355436218190051445, −14.16088787637230187063491974846, −13.80406668671687188565862172035, −12.44283347536795080852508920639, −11.67406881078655785443790951429, −11.00107151307603359919183636527, −10.15099871046011472269406652954, −9.11084037671101681043233432247, −8.45049353660813631930301548115, −7.89441798538954914878348973008, −6.89198290474100202357356870839, −6.22931240405347572719601468960, −5.82830709068435374614329407315, −4.20513001875261532982027203839, −2.823899531446915011459057309259, −2.05629233536373059110286850189, −1.545285413052898642548212127885, 0.34449320020325853539292125349, 1.51632590522295689575173667281, 2.47724358373499555779024118597, 3.597111865969676608548401273162, 4.33172403135543008502407640257, 5.13791075078436974544110310701, 6.32119605889315387113952686085, 7.58328846468625678069110880510, 8.27758563897807117498385508357, 8.778731849535016890700146914386, 9.64738850950233280428408574412, 10.33008831271534851621632206715, 10.96563897469509068709733428837, 11.74210954347897585959658103100, 12.81371387660699443588773343463, 13.52584973706260862078463957995, 14.37170933193263207213464190169, 15.54745771787330083186445990025, 16.01642083101032564673263384903, 16.73525806238906814178125394236, 17.46747594287599246904721951104, 17.93068767868959746710116249985, 19.18385710407717173813109712003, 19.99627606685463263302721332129, 20.4021543131727761765489034715

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