Properties

Degree $2$
Conductor $241$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.91·2-s − 0.186·3-s + 1.65·4-s − 2.25·5-s + 0.355·6-s + 3.52·7-s + 0.663·8-s − 2.96·9-s + 4.30·10-s + 0.515·11-s − 0.307·12-s − 5.38·13-s − 6.74·14-s + 0.419·15-s − 4.57·16-s − 4.16·17-s + 5.66·18-s + 4.92·19-s − 3.72·20-s − 0.657·21-s − 0.985·22-s − 7.69·23-s − 0.123·24-s + 0.0674·25-s + 10.2·26-s + 1.11·27-s + 5.83·28-s + ⋯
L(s)  = 1  − 1.35·2-s − 0.107·3-s + 0.826·4-s − 1.00·5-s + 0.145·6-s + 1.33·7-s + 0.234·8-s − 0.988·9-s + 1.36·10-s + 0.155·11-s − 0.0888·12-s − 1.49·13-s − 1.80·14-s + 0.108·15-s − 1.14·16-s − 1.01·17-s + 1.33·18-s + 1.13·19-s − 0.831·20-s − 0.143·21-s − 0.210·22-s − 1.60·23-s − 0.0252·24-s + 0.0134·25-s + 2.01·26-s + 0.213·27-s + 1.10·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 241 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 241 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(241\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{241} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 241,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad241 \( 1 + T \)
good2 \( 1 + 1.91T + 2T^{2} \)
3 \( 1 + 0.186T + 3T^{2} \)
5 \( 1 + 2.25T + 5T^{2} \)
7 \( 1 - 3.52T + 7T^{2} \)
11 \( 1 - 0.515T + 11T^{2} \)
13 \( 1 + 5.38T + 13T^{2} \)
17 \( 1 + 4.16T + 17T^{2} \)
19 \( 1 - 4.92T + 19T^{2} \)
23 \( 1 + 7.69T + 23T^{2} \)
29 \( 1 + 8.93T + 29T^{2} \)
31 \( 1 + 4.43T + 31T^{2} \)
37 \( 1 - 5.99T + 37T^{2} \)
41 \( 1 + 8.99T + 41T^{2} \)
43 \( 1 + 1.66T + 43T^{2} \)
47 \( 1 - 8.55T + 47T^{2} \)
53 \( 1 - 13.1T + 53T^{2} \)
59 \( 1 - 9.25T + 59T^{2} \)
61 \( 1 + 10.4T + 61T^{2} \)
67 \( 1 + 3.91T + 67T^{2} \)
71 \( 1 + 13.6T + 71T^{2} \)
73 \( 1 - 11.5T + 73T^{2} \)
79 \( 1 + 1.43T + 79T^{2} \)
83 \( 1 - 1.73T + 83T^{2} \)
89 \( 1 + 1.07T + 89T^{2} \)
97 \( 1 + 16.0T + 97T^{2} \)
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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

−11.63102528619926898386477463925, −10.70599727731048017273416349635, −9.550309840797062119224988995294, −8.614027947305890179654945538483, −7.79783467451283422206483284993, −7.29425027975600980879488973566, −5.38892382368832931887551820150, −4.19330997205883169168635222870, −2.11090369589586914936954379992, 0, 2.11090369589586914936954379992, 4.19330997205883169168635222870, 5.38892382368832931887551820150, 7.29425027975600980879488973566, 7.79783467451283422206483284993, 8.614027947305890179654945538483, 9.550309840797062119224988995294, 10.70599727731048017273416349635, 11.63102528619926898386477463925

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