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Properties

Label 1-41-41.5-r0-0-0
Degree $1$
Conductor $41$
Sign $-0.655 + 0.755i$
Analytic cond. $0.190403$
Root an. cond. $0.190403$
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  + (−0.309 + 0.951i)2-s + i·3-s + (−0.809 − 0.587i)4-s + (0.809 + 0.587i)5-s + (−0.951 − 0.309i)6-s + (−0.951 + 0.309i)7-s + (0.809 − 0.587i)8-s − 9-s + (−0.809 + 0.587i)10-s + (−0.587 − 0.809i)11-s + (0.587 − 0.809i)12-s + (0.951 + 0.309i)13-s − i·14-s + (−0.587 + 0.809i)15-s + (0.309 + 0.951i)16-s + (0.587 + 0.809i)17-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)2-s + i·3-s + (−0.809 − 0.587i)4-s + (0.809 + 0.587i)5-s + (−0.951 − 0.309i)6-s + (−0.951 + 0.309i)7-s + (0.809 − 0.587i)8-s − 9-s + (−0.809 + 0.587i)10-s + (−0.587 − 0.809i)11-s + (0.587 − 0.809i)12-s + (0.951 + 0.309i)13-s − i·14-s + (−0.587 + 0.809i)15-s + (0.309 + 0.951i)16-s + (0.587 + 0.809i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.655 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.655 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(41\)
Sign: $-0.655 + 0.755i$
Analytic conductor: \(0.190403\)
Root analytic conductor: \(0.190403\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{41} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 41,\ (0:\ ),\ -0.655 + 0.755i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2801465394 + 0.6140679299i\)
\(L(\frac12)\) \(\approx\) \(0.2801465394 + 0.6140679299i\)
\(L(1)\) \(\approx\) \(0.5734284390 + 0.5930784754i\)
\(L(1)\) \(\approx\) \(0.5734284390 + 0.5930784754i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad41 \( 1 \)
good2 \( 1 + (-0.309 + 0.951i)T \)
3 \( 1 + iT \)
5 \( 1 + (0.809 + 0.587i)T \)
7 \( 1 + (-0.951 + 0.309i)T \)
11 \( 1 + (-0.587 - 0.809i)T \)
13 \( 1 + (0.951 + 0.309i)T \)
17 \( 1 + (0.587 + 0.809i)T \)
19 \( 1 + (0.951 - 0.309i)T \)
23 \( 1 + (0.309 - 0.951i)T \)
29 \( 1 + (0.587 - 0.809i)T \)
31 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + (-0.309 + 0.951i)T \)
47 \( 1 + (-0.951 - 0.309i)T \)
53 \( 1 + (0.587 - 0.809i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 + (-0.309 - 0.951i)T \)
67 \( 1 + (-0.587 + 0.809i)T \)
71 \( 1 + (-0.587 - 0.809i)T \)
73 \( 1 - T \)
79 \( 1 + iT \)
83 \( 1 + T \)
89 \( 1 + (-0.951 + 0.309i)T \)
97 \( 1 + (-0.587 + 0.809i)T \)
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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

−35.10499712465477364478882841137, −33.21662129098313663714876993963, −31.81409447030255783063407005822, −30.76308811358956675316912752501, −29.41764876945132837923308916740, −29.02177821748148144851282305129, −27.86148237196102878845736423653, −25.90667801265537948540425788925, −25.33099074473880328668157702079, −23.466107129557049158224405727910, −22.490469972200212513340525143091, −20.774994440699133436502694271451, −20.00179940371107740404376310103, −18.56519687746421340343106449450, −17.7482527608795010094573371486, −16.433362993817159623006172175411, −13.74869379310405876926877926946, −13.09040093951213288360144021535, −12.02194551725321178771848145504, −10.22108090754782045716285017311, −9.0192831737352309040861565865, −7.39999568213681981700585625182, −5.4655917911899526929422153414, −3.06041427664153564793013201341, −1.34857500774761483938576310346, 3.35867272412095926255412025927, 5.46733437656588294673745640010, 6.41690861336468675899600929265, 8.5670110099376959655248098021, 9.73266949971198676834557270004, 10.73784504014365992401764247541, 13.35925917748663304424703050294, 14.48740474016265119437656759912, 15.81055311858843358961112221347, 16.587073960422393464522276125899, 18.06518674266593802981561387000, 19.22782559880980794809910227504, 21.206546615616489856207743499365, 22.23388401519221220898427292355, 23.24242981329882961712794569937, 25.015359806090105680779212099767, 26.13699374108928977701118560870, 26.509142735608356198585148198273, 28.18898348823732339010643080277, 29.001688081910453756059770939267, 31.06413253283394543169747496170, 32.48710318404475745830191210573, 32.88923986364105431309250173688, 34.192853249830842268679470450731, 34.95168888950004213199913083884

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