Properties

Label 2-1441-1.1-c1-0-78
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  − 1.26·2-s + 0.450·3-s − 0.412·4-s + 1.25·5-s − 0.568·6-s + 3.91·7-s + 3.03·8-s − 2.79·9-s − 1.58·10-s − 11-s − 0.185·12-s − 4.06·13-s − 4.93·14-s + 0.567·15-s − 3.00·16-s − 4.44·17-s + 3.52·18-s − 2.43·19-s − 0.519·20-s + 1.76·21-s + 1.26·22-s + 0.395·23-s + 1.37·24-s − 3.41·25-s + 5.12·26-s − 2.61·27-s − 1.61·28-s + ⋯
L(s)  = 1  − 0.890·2-s + 0.260·3-s − 0.206·4-s + 0.563·5-s − 0.231·6-s + 1.47·7-s + 1.07·8-s − 0.932·9-s − 0.501·10-s − 0.301·11-s − 0.0536·12-s − 1.12·13-s − 1.31·14-s + 0.146·15-s − 0.751·16-s − 1.07·17-s + 0.830·18-s − 0.558·19-s − 0.116·20-s + 0.385·21-s + 0.268·22-s + 0.0825·23-s + 0.279·24-s − 0.682·25-s + 1.00·26-s − 0.503·27-s − 0.305·28-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: Trivial
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 \( 1 + 1.26T + 2T^{2} \)
3 \( 1 - 0.450T + 3T^{2} \)
5 \( 1 - 1.25T + 5T^{2} \)
7 \( 1 - 3.91T + 7T^{2} \)
13 \( 1 + 4.06T + 13T^{2} \)
17 \( 1 + 4.44T + 17T^{2} \)
19 \( 1 + 2.43T + 19T^{2} \)
23 \( 1 - 0.395T + 23T^{2} \)
29 \( 1 + 2.73T + 29T^{2} \)
31 \( 1 + 2.17T + 31T^{2} \)
37 \( 1 + 3.95T + 37T^{2} \)
41 \( 1 + 8.35T + 41T^{2} \)
43 \( 1 - 7.41T + 43T^{2} \)
47 \( 1 - 9.76T + 47T^{2} \)
53 \( 1 + 10.2T + 53T^{2} \)
59 \( 1 - 12.7T + 59T^{2} \)
61 \( 1 + 6.22T + 61T^{2} \)
67 \( 1 - 5.60T + 67T^{2} \)
71 \( 1 + 8.78T + 71T^{2} \)
73 \( 1 - 0.203T + 73T^{2} \)
79 \( 1 - 5.68T + 79T^{2} \)
83 \( 1 + 15.4T + 83T^{2} \)
89 \( 1 + 5.80T + 89T^{2} \)
97 \( 1 + 9.76T + 97T^{2} \)
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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

−9.010625749205666388769936864388, −8.447711956815855121607187911091, −7.80479470016669688818809626741, −7.03508123837579838585232783316, −5.65427992986139318293523554339, −4.98291217720287534102514819787, −4.14044992859514901062243744924, −2.43590809624747968167459460364, −1.75101713406692439030949905932, 0, 1.75101713406692439030949905932, 2.43590809624747968167459460364, 4.14044992859514901062243744924, 4.98291217720287534102514819787, 5.65427992986139318293523554339, 7.03508123837579838585232783316, 7.80479470016669688818809626741, 8.447711956815855121607187911091, 9.010625749205666388769936864388

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