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Properties

Label 1-41-41.39-r0-0-0
Degree $1$
Conductor $41$
Sign $0.998 - 0.0546i$
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  + (0.809 − 0.587i)2-s + i·3-s + (0.309 − 0.951i)4-s + (−0.309 + 0.951i)5-s + (0.587 + 0.809i)6-s + (0.587 − 0.809i)7-s + (−0.309 − 0.951i)8-s − 9-s + (0.309 + 0.951i)10-s + (−0.951 + 0.309i)11-s + (0.951 + 0.309i)12-s + (−0.587 − 0.809i)13-s − i·14-s + (−0.951 − 0.309i)15-s + (−0.809 − 0.587i)16-s + (0.951 − 0.309i)17-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)2-s + i·3-s + (0.309 − 0.951i)4-s + (−0.309 + 0.951i)5-s + (0.587 + 0.809i)6-s + (0.587 − 0.809i)7-s + (−0.309 − 0.951i)8-s − 9-s + (0.309 + 0.951i)10-s + (−0.951 + 0.309i)11-s + (0.951 + 0.309i)12-s + (−0.587 − 0.809i)13-s − i·14-s + (−0.951 − 0.309i)15-s + (−0.809 − 0.587i)16-s + (0.951 − 0.309i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0546i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0546i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(41\)
Sign: $0.998 - 0.0546i$
Analytic conductor: \(0.190403\)
Root analytic conductor: \(0.190403\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{41} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 41,\ (0:\ ),\ 0.998 - 0.0546i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.129671936 + 0.03090479340i\)
\(L(\frac12)\) \(\approx\) \(1.129671936 + 0.03090479340i\)
\(L(1)\) \(\approx\) \(1.322227822 - 0.05315054433i\)
\(L(1)\) \(\approx\) \(1.322227822 - 0.05315054433i\)

Euler product

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

−34.59520639187795716648458831967, −34.16861043989812346453272783705, −32.238153592637866111444509819200, −31.489405402067231122250440561727, −30.617599415532610430695920428076, −29.23893862630413733757048946241, −28.07951963043614213256338812964, −26.186317938482763892070729589088, −24.94230206140595690371319819610, −24.076376144214129182219007818202, −23.502572150618079892721104144953, −21.7323639414123143039126944890, −20.631855869560859961167345688865, −19.041518795132878748733061054050, −17.57513399349488813050433760686, −16.39393093867877940859434979673, −14.95960276829490043103993934336, −13.599743934719634620683617219086, −12.44473522972768681997404233210, −11.67276422692269033111284644861, −8.59528220814246365435159930406, −7.74970633197436859314584895425, −5.99843561868967454950909876860, −4.75503995342724152263230133857, −2.3896099792242055056464489347, 2.83549661438486924295403404285, 4.153844560458008011082015556947, 5.55758989764977514478673797676, 7.62346208459804067695605070925, 10.226540507657815418326248392470, 10.57283904518370061576740709234, 12.07723439694455812198762683418, 14.01761857520856659922657411531, 14.82124548405411017722416087920, 15.96490496088538480640062450427, 17.85964300699820506527126776579, 19.52894970343819910080929969423, 20.648839496803052459288957707159, 21.59018410696273392966021881723, 22.87907428205046598123193908177, 23.48843059638646099130745566060, 25.49399055319717123963360836233, 26.9448981892083700924816784731, 27.70191738172030685432953665523, 29.285260558466007719562981506954, 30.31158651926723552602355610799, 31.44548267406521078325543916966, 32.44120319742411785497274380577, 33.85871596824283451829302680694, 34.08067965291981728606114944831

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