Properties

Label 1-1441-1441.190-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: $\chi_{1441} (190, \cdot )$
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

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

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