Properties

Label 1-41-41.37-r0-0-0
Degree $1$
Conductor $41$
Sign $0.278 - 0.960i$
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 + 3-s + (−0.809 − 0.587i)4-s + (−0.809 − 0.587i)5-s + (0.309 − 0.951i)6-s + (0.309 + 0.951i)7-s + (−0.809 + 0.587i)8-s + 9-s + (−0.809 + 0.587i)10-s + (−0.809 + 0.587i)11-s + (−0.809 − 0.587i)12-s + (0.309 − 0.951i)13-s + 14-s + (−0.809 − 0.587i)15-s + (0.309 + 0.951i)16-s + (−0.809 + 0.587i)17-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)2-s + 3-s + (−0.809 − 0.587i)4-s + (−0.809 − 0.587i)5-s + (0.309 − 0.951i)6-s + (0.309 + 0.951i)7-s + (−0.809 + 0.587i)8-s + 9-s + (−0.809 + 0.587i)10-s + (−0.809 + 0.587i)11-s + (−0.809 − 0.587i)12-s + (0.309 − 0.951i)13-s + 14-s + (−0.809 − 0.587i)15-s + (0.309 + 0.951i)16-s + (−0.809 + 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.278 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.278 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(41\)
Sign: $0.278 - 0.960i$
Analytic conductor: \(0.190403\)
Root analytic conductor: \(0.190403\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{41} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 41,\ (0:\ ),\ 0.278 - 0.960i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8250835934 - 0.6197106480i\)
\(L(\frac12)\) \(\approx\) \(0.8250835934 - 0.6197106480i\)
\(L(1)\) \(\approx\) \(1.070122344 - 0.5730171816i\)
\(L(1)\) \(\approx\) \(1.070122344 - 0.5730171816i\)

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 \( 1 + T \)
5 \( 1 + (-0.809 - 0.587i)T \)
7 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (-0.809 + 0.587i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (0.309 - 0.951i)T \)
29 \( 1 + (-0.809 - 0.587i)T \)
31 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + (0.309 - 0.951i)T \)
47 \( 1 + (0.309 - 0.951i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 + (0.309 + 0.951i)T \)
67 \( 1 + (-0.809 - 0.587i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + T \)
79 \( 1 + T \)
83 \( 1 + T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (-0.809 - 0.587i)T \)
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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

−35.19008638890914603095038273837, −33.82839128167056867325456079825, −32.92139352281183083444865848184, −31.45530444284392904624324401250, −31.03771824553998757920428658795, −29.76718453763661123670897539128, −27.390443716364159612748828770170, −26.4792892378609829337913806344, −25.96557563167173505311529639944, −24.16571144602850920042811840121, −23.65576375375799380294952238126, −22.07149708508442004534531952067, −20.69132650065418765184359524627, −19.24183438815843948475154732417, −18.080281822858437749555795813057, −16.26108551880828914915638095581, −15.31953014515352981213066479377, −14.08325133390599986554445888768, −13.27882906686200459805248408587, −11.13132987444381347696673682061, −9.14411943828619609734940737642, −7.75516509557503879421411376562, −6.97282782628456534474292566694, −4.51859688769630040817601833802, −3.27335771390723944088657615734, 2.100722921310850653829488116806, 3.67414666270539833092263815047, 5.16155679606880506323536985949, 8.03708858532477549760473107711, 8.97723727364119518580192942063, 10.58157050300445538714198743777, 12.308918533699409222837408734, 13.04744359379762061864766826571, 14.7926972566912720234549092481, 15.60950939645293157543404890661, 18.10950230386569969369228354517, 19.11654984157057078849662208101, 20.349266191872907233856881087305, 20.92628166471918794084273713911, 22.465218645566179261673298817113, 23.890308178930650468436750847845, 25.003556879979600158727177359147, 26.67361747233246764109943059668, 27.75524187309585326196462338515, 28.69223933172473169492365702264, 30.430052857506175411723987451393, 31.18295738436761884860698625972, 31.84539120858723620403879194171, 33.01560469462717730494608293525, 35.07327276490164370548929553030

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