Properties

Label 2-41-1.1-c1-0-2
Degree $2$
Conductor $41$
Sign $1$
Analytic cond. $0.327386$
Root an. cond. $0.572177$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.90·2-s − 2.21·3-s + 1.62·4-s − 2.90·5-s − 4.21·6-s + 4.21·7-s − 0.719·8-s + 1.90·9-s − 5.52·10-s + 2.83·11-s − 3.59·12-s − 0.622·13-s + 8.02·14-s + 6.42·15-s − 4.61·16-s − 2·17-s + 3.62·18-s − 0.836·19-s − 4.70·20-s − 9.33·21-s + 5.39·22-s − 3.05·23-s + 1.59·24-s + 3.42·25-s − 1.18·26-s + 2.42·27-s + 6.83·28-s + ⋯
L(s)  = 1  + 1.34·2-s − 1.27·3-s + 0.811·4-s − 1.29·5-s − 1.72·6-s + 1.59·7-s − 0.254·8-s + 0.634·9-s − 1.74·10-s + 0.855·11-s − 1.03·12-s − 0.172·13-s + 2.14·14-s + 1.65·15-s − 1.15·16-s − 0.485·17-s + 0.853·18-s − 0.191·19-s − 1.05·20-s − 2.03·21-s + 1.15·22-s − 0.636·23-s + 0.324·24-s + 0.685·25-s − 0.232·26-s + 0.467·27-s + 1.29·28-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(41\)
Sign: $1$
Analytic conductor: \(0.327386\)
Root analytic conductor: \(0.572177\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 41,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9440427937\)
\(L(\frac12)\) \(\approx\) \(0.9440427937\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad41 \( 1 - T \)
good2 \( 1 - 1.90T + 2T^{2} \)
3 \( 1 + 2.21T + 3T^{2} \)
5 \( 1 + 2.90T + 5T^{2} \)
7 \( 1 - 4.21T + 7T^{2} \)
11 \( 1 - 2.83T + 11T^{2} \)
13 \( 1 + 0.622T + 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + 0.836T + 19T^{2} \)
23 \( 1 + 3.05T + 23T^{2} \)
29 \( 1 - 2.42T + 29T^{2} \)
31 \( 1 - 9.80T + 31T^{2} \)
37 \( 1 + 8.70T + 37T^{2} \)
43 \( 1 + 1.37T + 43T^{2} \)
47 \( 1 + 6.77T + 47T^{2} \)
53 \( 1 - 6.42T + 53T^{2} \)
59 \( 1 + 7.05T + 59T^{2} \)
61 \( 1 - 5.18T + 61T^{2} \)
67 \( 1 + 2.83T + 67T^{2} \)
71 \( 1 - 8.96T + 71T^{2} \)
73 \( 1 - 1.39T + 73T^{2} \)
79 \( 1 - 8.40T + 79T^{2} \)
83 \( 1 - 8.85T + 83T^{2} \)
89 \( 1 + 6.29T + 89T^{2} \)
97 \( 1 + 6.85T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.83554185992882929850322167910, −14.92936903700120757901000297592, −13.92225426122259878414512191078, −12.04937815819870602134998117258, −11.86952377845177227169703056383, −10.94269555113269070169198308114, −8.308711178390771027808150974372, −6.61041296440600507604659986604, −5.05968586475281254475232400968, −4.19259081075605077803043927645, 4.19259081075605077803043927645, 5.05968586475281254475232400968, 6.61041296440600507604659986604, 8.308711178390771027808150974372, 10.94269555113269070169198308114, 11.86952377845177227169703056383, 12.04937815819870602134998117258, 13.92225426122259878414512191078, 14.92936903700120757901000297592, 15.83554185992882929850322167910

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