Properties

Label 2-1441-1.1-c1-0-40
Degree $2$
Conductor $1441$
Sign $-1$
Analytic cond. $11.5064$
Root an. cond. $3.39211$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.169·2-s − 1.85·3-s − 1.97·4-s − 3.73·5-s + 0.313·6-s + 1.69·7-s + 0.673·8-s + 0.423·9-s + 0.633·10-s + 11-s + 3.64·12-s + 0.706·13-s − 0.287·14-s + 6.90·15-s + 3.82·16-s − 0.521·17-s − 0.0718·18-s + 5.59·19-s + 7.35·20-s − 3.13·21-s − 0.169·22-s + 3.04·23-s − 1.24·24-s + 8.92·25-s − 0.119·26-s + 4.76·27-s − 3.34·28-s + ⋯
L(s)  = 1  − 0.119·2-s − 1.06·3-s − 0.985·4-s − 1.66·5-s + 0.128·6-s + 0.640·7-s + 0.238·8-s + 0.141·9-s + 0.200·10-s + 0.301·11-s + 1.05·12-s + 0.195·13-s − 0.0768·14-s + 1.78·15-s + 0.957·16-s − 0.126·17-s − 0.0169·18-s + 1.28·19-s + 1.64·20-s − 0.684·21-s − 0.0361·22-s + 0.635·23-s − 0.254·24-s + 1.78·25-s − 0.0235·26-s + 0.917·27-s − 0.631·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-1$
Analytic conductor: \(11.5064\)
Root analytic conductor: \(3.39211\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1441,\ (\ :1/2),\ -1)\)

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)$
bad11 \( 1 - T \)
131 \( 1 - T \)
good2 \( 1 + 0.169T + 2T^{2} \)
3 \( 1 + 1.85T + 3T^{2} \)
5 \( 1 + 3.73T + 5T^{2} \)
7 \( 1 - 1.69T + 7T^{2} \)
13 \( 1 - 0.706T + 13T^{2} \)
17 \( 1 + 0.521T + 17T^{2} \)
19 \( 1 - 5.59T + 19T^{2} \)
23 \( 1 - 3.04T + 23T^{2} \)
29 \( 1 + 8.02T + 29T^{2} \)
31 \( 1 - 1.55T + 31T^{2} \)
37 \( 1 + 7.66T + 37T^{2} \)
41 \( 1 + 5.80T + 41T^{2} \)
43 \( 1 - 5.42T + 43T^{2} \)
47 \( 1 + 9.01T + 47T^{2} \)
53 \( 1 - 7.15T + 53T^{2} \)
59 \( 1 - 15.1T + 59T^{2} \)
61 \( 1 - 9.76T + 61T^{2} \)
67 \( 1 - 3.25T + 67T^{2} \)
71 \( 1 + 10.0T + 71T^{2} \)
73 \( 1 + 9.77T + 73T^{2} \)
79 \( 1 - 10.7T + 79T^{2} \)
83 \( 1 + 6.21T + 83T^{2} \)
89 \( 1 + 2.41T + 89T^{2} \)
97 \( 1 - 15.0T + 97T^{2} \)
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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

−8.926477517354261275367763250976, −8.356064294107203512695907323827, −7.56566182829985000862675259092, −6.83765300207745913498283287435, −5.44148624827543208396205307339, −5.03078566412464671108981201021, −4.07830299886595208101077415089, −3.36404343138609451061666138186, −1.10166669474520390967879144631, 0, 1.10166669474520390967879144631, 3.36404343138609451061666138186, 4.07830299886595208101077415089, 5.03078566412464671108981201021, 5.44148624827543208396205307339, 6.83765300207745913498283287435, 7.56566182829985000862675259092, 8.356064294107203512695907323827, 8.926477517354261275367763250976

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