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Properties

Label 1-41-41.24-r1-0-0
Degree $1$
Conductor $41$
Sign $0.848 - 0.529i$
Analytic cond. $4.40606$
Root an. cond. $4.40606$
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  + (−0.951 + 0.309i)2-s + (0.707 − 0.707i)3-s + (0.809 − 0.587i)4-s + (0.587 + 0.809i)5-s + (−0.453 + 0.891i)6-s + (−0.453 − 0.891i)7-s + (−0.587 + 0.809i)8-s − i·9-s + (−0.809 − 0.587i)10-s + (0.987 − 0.156i)11-s + (0.156 − 0.987i)12-s + (0.891 + 0.453i)13-s + (0.707 + 0.707i)14-s + (0.987 + 0.156i)15-s + (0.309 − 0.951i)16-s + (−0.156 − 0.987i)17-s + ⋯
L(s)  = 1  + (−0.951 + 0.309i)2-s + (0.707 − 0.707i)3-s + (0.809 − 0.587i)4-s + (0.587 + 0.809i)5-s + (−0.453 + 0.891i)6-s + (−0.453 − 0.891i)7-s + (−0.587 + 0.809i)8-s − i·9-s + (−0.809 − 0.587i)10-s + (0.987 − 0.156i)11-s + (0.156 − 0.987i)12-s + (0.891 + 0.453i)13-s + (0.707 + 0.707i)14-s + (0.987 + 0.156i)15-s + (0.309 − 0.951i)16-s + (−0.156 − 0.987i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.848 - 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.848 - 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(41\)
Sign: $0.848 - 0.529i$
Analytic conductor: \(4.40606\)
Root analytic conductor: \(4.40606\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{41} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 41,\ (1:\ ),\ 0.848 - 0.529i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.290662765 - 0.3701519624i\)
\(L(\frac12)\) \(\approx\) \(1.290662765 - 0.3701519624i\)
\(L(1)\) \(\approx\) \(1.012857173 - 0.1422301524i\)
\(L(1)\) \(\approx\) \(1.012857173 - 0.1422301524i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad41 \( 1 \)
good2 \( 1 + (-0.951 + 0.309i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (0.587 + 0.809i)T \)
7 \( 1 + (-0.453 - 0.891i)T \)
11 \( 1 + (0.987 - 0.156i)T \)
13 \( 1 + (0.891 + 0.453i)T \)
17 \( 1 + (-0.156 - 0.987i)T \)
19 \( 1 + (0.891 - 0.453i)T \)
23 \( 1 + (-0.309 - 0.951i)T \)
29 \( 1 + (-0.156 + 0.987i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + (-0.951 + 0.309i)T \)
47 \( 1 + (-0.453 + 0.891i)T \)
53 \( 1 + (0.156 - 0.987i)T \)
59 \( 1 + (0.309 + 0.951i)T \)
61 \( 1 + (0.951 + 0.309i)T \)
67 \( 1 + (-0.987 - 0.156i)T \)
71 \( 1 + (-0.987 + 0.156i)T \)
73 \( 1 + iT \)
79 \( 1 + (-0.707 + 0.707i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.453 - 0.891i)T \)
97 \( 1 + (0.987 + 0.156i)T \)
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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

−35.16953826672849825713142245023, −33.432371681972833646364191817376, −32.54318920974087497670573418962, −31.22358903506587516317489531311, −29.90094082546060090528705390120, −28.26539694758653802301110025990, −27.970053666244520938101695922133, −26.410109986367595022836196426512, −25.30722625520054629005852998599, −24.76910259127585905367857608026, −22.13715886980792255520170861309, −21.15512074413777948663577058459, −20.14887540479150465917403652356, −19.09390058413052254190522190904, −17.528310374078353587244749186453, −16.29019247424876725496171629331, −15.28297206918018735515132071371, −13.36314981160954994677077047186, −11.86518001632856217848663568366, −10.053922552048582697745725065856, −9.16316079866680465358531085353, −8.24100364395326289344355177591, −5.92404118713321510093341496873, −3.61032993913576862797806004036, −1.80188429269940173261359849970, 1.26894083228531502524611001975, 3.09815583481187596222012844934, 6.48646759412420371961568334669, 7.09219651136961143817733150443, 8.820203933389740198737255571132, 9.97954939783475929784352693553, 11.511014200418459766631518851960, 13.67017727872615654664489330563, 14.46010629549467334223453887126, 16.182075520493339104463611001855, 17.64592451811580798214800780017, 18.57471289465196283960179581962, 19.66051911547850862767150655000, 20.702541779249651629529360199696, 22.72659398109181290088627582376, 24.16747329122293174061373553719, 25.2936763501737332318363206736, 26.160742140059396429139163590662, 26.95811455858225637514130320366, 28.8382248671874333482123579156, 29.765372238410249712060042020826, 30.58271310939852356773442304456, 32.57043249536803759301541043769, 33.406744505503294891739428832817, 34.88209541341057386016046961266

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