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Properties

Label 1-41e2-1681.42-r0-0-0
Degree $1$
Conductor $1681$
Sign $0.157 - 0.987i$
Analytic cond. $7.80653$
Root an. cond. $7.80653$
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  + (0.953 − 0.301i)2-s + (−0.771 + 0.636i)3-s + (0.817 − 0.575i)4-s + (0.896 − 0.443i)5-s + (−0.543 + 0.839i)6-s + (0.720 − 0.693i)7-s + (0.606 − 0.795i)8-s + (0.190 − 0.981i)9-s + (0.720 − 0.693i)10-s + (−0.927 − 0.373i)11-s + (−0.264 + 0.964i)12-s + (0.953 − 0.301i)13-s + (0.477 − 0.878i)14-s + (−0.409 + 0.912i)15-s + (0.338 − 0.941i)16-s + (−0.771 − 0.636i)17-s + ⋯
L(s)  = 1  + (0.953 − 0.301i)2-s + (−0.771 + 0.636i)3-s + (0.817 − 0.575i)4-s + (0.896 − 0.443i)5-s + (−0.543 + 0.839i)6-s + (0.720 − 0.693i)7-s + (0.606 − 0.795i)8-s + (0.190 − 0.981i)9-s + (0.720 − 0.693i)10-s + (−0.927 − 0.373i)11-s + (−0.264 + 0.964i)12-s + (0.953 − 0.301i)13-s + (0.477 − 0.878i)14-s + (−0.409 + 0.912i)15-s + (0.338 − 0.941i)16-s + (−0.771 − 0.636i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1681 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.157 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1681 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.157 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1681\)    =    \(41^{2}\)
Sign: $0.157 - 0.987i$
Analytic conductor: \(7.80653\)
Root analytic conductor: \(7.80653\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1681} (42, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1681,\ (0:\ ),\ 0.157 - 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.200423995 - 1.877739987i\)
\(L(\frac12)\) \(\approx\) \(2.200423995 - 1.877739987i\)
\(L(1)\) \(\approx\) \(1.730940166 - 0.5988062016i\)
\(L(1)\) \(\approx\) \(1.730940166 - 0.5988062016i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad41 \( 1 \)
good2 \( 1 + (0.953 - 0.301i)T \)
3 \( 1 + (-0.771 + 0.636i)T \)
5 \( 1 + (0.896 - 0.443i)T \)
7 \( 1 + (0.720 - 0.693i)T \)
11 \( 1 + (-0.927 - 0.373i)T \)
13 \( 1 + (0.953 - 0.301i)T \)
17 \( 1 + (-0.771 - 0.636i)T \)
19 \( 1 + (-0.665 + 0.746i)T \)
23 \( 1 + (0.190 + 0.981i)T \)
29 \( 1 + (0.988 + 0.152i)T \)
31 \( 1 + (0.0383 - 0.999i)T \)
37 \( 1 + (0.817 + 0.575i)T \)
43 \( 1 + (-0.927 - 0.373i)T \)
47 \( 1 + (0.720 - 0.693i)T \)
53 \( 1 + (-0.927 - 0.373i)T \)
59 \( 1 + (0.0383 + 0.999i)T \)
61 \( 1 + (-0.771 + 0.636i)T \)
67 \( 1 + (-0.997 + 0.0765i)T \)
71 \( 1 + (0.988 - 0.152i)T \)
73 \( 1 + (-0.114 - 0.993i)T \)
79 \( 1 + (0.0383 + 0.999i)T \)
83 \( 1 + (-0.927 + 0.373i)T \)
89 \( 1 + (-0.997 - 0.0765i)T \)
97 \( 1 + (0.606 + 0.795i)T \)
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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

−20.94926351778372049572108946745, −19.88719833352257626080503624453, −18.7807122956725485150472531872, −18.11098786029329902319000857014, −17.59783480453386798866475753307, −16.92657255950312712499166277535, −15.86715669180409942898931753217, −15.35836148177514631182431996585, −14.363637687221641938586141246031, −13.79017586871112917302664469304, −12.78946673747027796340659528588, −12.72462408354298556865913895479, −11.45383133361225447068281507837, −10.92435617511754109480314321251, −10.39277328775652321597987062181, −8.79745591285466570161687675863, −8.12578712809297895720496208086, −7.09921070789243118305634998035, −6.308724606866337779652178526481, −5.99280567802380737809589205755, −4.92327439916229471031643645417, −4.548270526863826790485284196078, −2.873396373825008127259810798592, −2.2199566656763253468610080989, −1.5195965528927003037100248976, 0.81412763026561224288125991284, 1.68412540028670547747161955581, 2.840757418355041239374349263329, 3.90430929926967759257621607417, 4.637378764601174262588397568771, 5.28745138851736352301311612800, 5.947347824972603762018346794006, 6.64347611008819788024774828015, 7.820311417541897855799129485025, 8.88725217053783771188460004161, 10.017631847980387014201485021328, 10.458754675883201682995776455824, 11.14879746553393002040087699458, 11.78046614458375789414057340051, 12.85516877521384920776173622839, 13.45591167773322946108438941234, 13.96224436451685221698853123997, 15.056161116705376336142120393476, 15.633705049316739179344416532740, 16.492263887800656930218795095575, 16.992923627101117952410587073713, 17.99564377401876502056517426271, 18.46226770221293883297082061301, 19.90269128866860735945331197200, 20.59757970225541141712664768155

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