Properties

Label 1-41-41.40-r0-0-0
Degree $1$
Conductor $41$
Sign $1$
Analytic cond. $0.190403$
Root an. cond. $0.190403$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s + 5-s − 6-s − 7-s + 8-s + 9-s + 10-s − 11-s − 12-s − 13-s − 14-s − 15-s + 16-s − 17-s + 18-s − 19-s + 20-s + 21-s − 22-s + 23-s − 24-s + 25-s − 26-s − 27-s − 28-s + ⋯
L(s)  = 1  + 2-s − 3-s + 4-s + 5-s − 6-s − 7-s + 8-s + 9-s + 10-s − 11-s − 12-s − 13-s − 14-s − 15-s + 16-s − 17-s + 18-s − 19-s + 20-s + 21-s − 22-s + 23-s − 24-s + 25-s − 26-s − 27-s − 28-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(41\)
Sign: $1$
Analytic conductor: \(0.190403\)
Root analytic conductor: \(0.190403\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{41} (40, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 41,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.107354323\)
\(L(\frac12)\) \(\approx\) \(1.107354323\)
\(L(1)\) \(\approx\) \(1.299093061\)
\(L(1)\) \(\approx\) \(1.299093061\)

Euler product

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

−34.47546215947206763426136146568, −33.6661521839196955099062132230, −32.69373074221944522676079722491, −31.65407261004172014783956532469, −29.93039824682136545224343416482, −29.1356468523485230293113881357, −28.541212871826520343874760780792, −26.39663799101612925813311486865, −25.09402232783170755205407028706, −23.946754832260061054858339580998, −22.71883368285344400541134984112, −21.93785421040218099753139884926, −20.91174866242924281026996303425, −19.13793575707422622348777862662, −17.44547462637966107901631736435, −16.3928748224285834048766910693, −15.1131494584028990933990985924, −13.22815980831450135879449383492, −12.71210543720683323663551991250, −10.969009693594641705466479713, −9.86659424108116642926443752381, −6.97434799988083841026559542165, −5.94119908387877980482735297655, −4.721950906732446356931143772002, −2.47534046077300626035956606785, 2.47534046077300626035956606785, 4.721950906732446356931143772002, 5.94119908387877980482735297655, 6.97434799988083841026559542165, 9.86659424108116642926443752381, 10.969009693594641705466479713, 12.71210543720683323663551991250, 13.22815980831450135879449383492, 15.1131494584028990933990985924, 16.3928748224285834048766910693, 17.44547462637966107901631736435, 19.13793575707422622348777862662, 20.91174866242924281026996303425, 21.93785421040218099753139884926, 22.71883368285344400541134984112, 23.946754832260061054858339580998, 25.09402232783170755205407028706, 26.39663799101612925813311486865, 28.541212871826520343874760780792, 29.1356468523485230293113881357, 29.93039824682136545224343416482, 31.65407261004172014783956532469, 32.69373074221944522676079722491, 33.6661521839196955099062132230, 34.47546215947206763426136146568

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