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Properties

Label 1-41-41.16-r0-0-0
Degree $1$
Conductor $41$
Sign $0.995 - 0.0984i$
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 + 3-s + (0.309 + 0.951i)4-s + (0.309 + 0.951i)5-s + (−0.809 − 0.587i)6-s + (−0.809 + 0.587i)7-s + (0.309 − 0.951i)8-s + 9-s + (0.309 − 0.951i)10-s + (0.309 − 0.951i)11-s + (0.309 + 0.951i)12-s + (−0.809 − 0.587i)13-s + 14-s + (0.309 + 0.951i)15-s + (−0.809 + 0.587i)16-s + (0.309 − 0.951i)17-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + 3-s + (0.309 + 0.951i)4-s + (0.309 + 0.951i)5-s + (−0.809 − 0.587i)6-s + (−0.809 + 0.587i)7-s + (0.309 − 0.951i)8-s + 9-s + (0.309 − 0.951i)10-s + (0.309 − 0.951i)11-s + (0.309 + 0.951i)12-s + (−0.809 − 0.587i)13-s + 14-s + (0.309 + 0.951i)15-s + (−0.809 + 0.587i)16-s + (0.309 − 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(41\)
Sign: $0.995 - 0.0984i$
Analytic conductor: \(0.190403\)
Root analytic conductor: \(0.190403\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{41} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 41,\ (0:\ ),\ 0.995 - 0.0984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7455334340 - 0.03677912394i\)
\(L(\frac12)\) \(\approx\) \(0.7455334340 - 0.03677912394i\)
\(L(1)\) \(\approx\) \(0.8900534876 - 0.06674609991i\)
\(L(1)\) \(\approx\) \(0.8900534876 - 0.06674609991i\)

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

−35.43132532812880055904054468908, −33.651629970204183219924266973036, −32.51832370239008534271580035147, −31.993385189589146769933691175097, −30.12923083753853189220502938894, −28.8173137731883130249614858930, −27.7047972898428264478689921645, −26.3256097145118734037404244071, −25.59268355578134950437052162530, −24.55986181159426414904549827827, −23.443200348798631786061678750935, −21.31619974259763167362692252594, −19.79598565107287318779227136891, −19.530511955587848668255457670827, −17.59286087239545414201747580736, −16.53392215373349275336315384235, −15.27066277231698947437112824735, −13.96039178078254166781491837325, −12.582096410128919708405154225911, −10.027702356974310706584801250, −9.33152964779482509999495616333, −7.96882447807059451896091604958, −6.611410707952482682615139206101, −4.4408547695280499784099535573, −1.86555292245387372042768418810, 2.44639998913664679387411711450, 3.37986815129714169246954255334, 6.59472672592148469668273757397, 8.07781246072026666106026142545, 9.444870272057090076828322463624, 10.38770629051250430729559448267, 12.16985736728268508988053770719, 13.614787585126604580686748703626, 15.07647732172535488466971997971, 16.50671515930099050466109880601, 18.3553921596330599523539894136, 18.99159084048558910991313848803, 20.06622376342121561691408690861, 21.52472984746543320073405784006, 22.33248880126053757693937610859, 24.819153959945695929765308458227, 25.61659851577049008081998688502, 26.61450235650097264445501321489, 27.52842023521782604031147208607, 29.41642241216399810957501809149, 29.85242469011908637914916375622, 31.27956607361527267355543679985, 32.26063297660022500019268617853, 34.10669650933816554102109518155, 35.102600872379379034127648368092

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