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Properties

Label 1-41-41.2-r0-0-0
Degree $1$
Conductor $41$
Sign $0.297 - 0.954i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.297 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.297 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(41\)
Sign: $0.297 - 0.954i$
Analytic conductor: \(0.190403\)
Root analytic conductor: \(0.190403\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{41} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 41,\ (0:\ ),\ 0.297 - 0.954i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8928840714 - 0.6570267233i\)
\(L(\frac12)\) \(\approx\) \(0.8928840714 - 0.6570267233i\)
\(L(1)\) \(\approx\) \(1.154853410 - 0.5989474567i\)
\(L(1)\) \(\approx\) \(1.154853410 - 0.5989474567i\)

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

−35.139605886665011891542894174165, −33.46746468988139399551548205095, −32.782264608753106317951239389613, −32.06960522527543339386207832608, −30.87250889760960142097538974598, −29.372629557053296332903476055077, −27.89296849863771461471009541777, −26.74855378775485749123975512401, −25.56721072540963115339447807669, −24.36626610625176020449402040024, −22.92509956413567631018660006860, −22.26809236999670496068729801821, −20.50569673847266237240268839597, −20.189197204386315098107499683730, −17.30219971276130379306458995832, −16.40290194090604759291620231081, −15.55211626616948348757573474136, −14.10619413649570521421724221429, −12.79505015039196794519074475876, −11.339291492537046572840576796046, −9.49656832298830935138564509652, −8.03058837244419362719445192743, −6.11492251627196633975151480423, −4.543044347580638406677220510781, −3.61235238636611322277255481423, 2.12833102826280710218526300898, 3.57366127330464903853511056556, 6.01964263880981140984671889665, 6.87482551716963960156762497899, 9.135822946699873899647360707310, 11.19737683867800183305950448029, 11.94546368102915107242660114156, 13.39370160693107082569299162048, 14.41472864435699897739947345948, 15.77500181988167621211055406445, 18.04150959896894170148699016939, 19.118391028977642381192585469213, 19.77446348792423345635446136098, 21.80822144904119064614744579094, 22.55248036125406956426520097660, 23.75790317084382506359138361710, 24.83363566024090379994327202211, 26.16158071979735830484079850639, 28.1057073592349070702758750373, 29.090393401725065972443071247369, 30.241002661010403263797446384058, 30.89673516830975951010523076133, 31.96990005380781094914519488794, 33.50071590821009769511061212157, 34.743805665274508346415135787993

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