Properties

Label 2-1441-1.1-c1-0-91
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  + 0.338·2-s + 0.415·3-s − 1.88·4-s + 2.48·5-s + 0.140·6-s + 1.37·7-s − 1.31·8-s − 2.82·9-s + 0.842·10-s + 11-s − 0.782·12-s − 6.16·13-s + 0.464·14-s + 1.03·15-s + 3.32·16-s − 6.22·17-s − 0.958·18-s − 1.56·19-s − 4.68·20-s + 0.569·21-s + 0.338·22-s − 8.96·23-s − 0.546·24-s + 1.18·25-s − 2.09·26-s − 2.41·27-s − 2.58·28-s + ⋯
L(s)  = 1  + 0.239·2-s + 0.239·3-s − 0.942·4-s + 1.11·5-s + 0.0574·6-s + 0.518·7-s − 0.465·8-s − 0.942·9-s + 0.266·10-s + 0.301·11-s − 0.225·12-s − 1.71·13-s + 0.124·14-s + 0.266·15-s + 0.830·16-s − 1.50·17-s − 0.225·18-s − 0.359·19-s − 1.04·20-s + 0.124·21-s + 0.0722·22-s − 1.87·23-s − 0.111·24-s + 0.236·25-s − 0.410·26-s − 0.465·27-s − 0.488·28-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 \( 1 - 0.338T + 2T^{2} \)
3 \( 1 - 0.415T + 3T^{2} \)
5 \( 1 - 2.48T + 5T^{2} \)
7 \( 1 - 1.37T + 7T^{2} \)
13 \( 1 + 6.16T + 13T^{2} \)
17 \( 1 + 6.22T + 17T^{2} \)
19 \( 1 + 1.56T + 19T^{2} \)
23 \( 1 + 8.96T + 23T^{2} \)
29 \( 1 - 0.948T + 29T^{2} \)
31 \( 1 - 9.67T + 31T^{2} \)
37 \( 1 - 2.06T + 37T^{2} \)
41 \( 1 + 9.20T + 41T^{2} \)
43 \( 1 - 1.99T + 43T^{2} \)
47 \( 1 - 5.37T + 47T^{2} \)
53 \( 1 - 11.2T + 53T^{2} \)
59 \( 1 + 7.99T + 59T^{2} \)
61 \( 1 - 1.55T + 61T^{2} \)
67 \( 1 + 15.1T + 67T^{2} \)
71 \( 1 + 0.208T + 71T^{2} \)
73 \( 1 + 14.1T + 73T^{2} \)
79 \( 1 - 0.383T + 79T^{2} \)
83 \( 1 - 2.50T + 83T^{2} \)
89 \( 1 + 5.72T + 89T^{2} \)
97 \( 1 - 7.59T + 97T^{2} \)
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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

−9.061811684785377215962391006203, −8.540145264607385820885226966358, −7.68838829936442250962379923547, −6.40684021302317291689859230052, −5.77117588712804088861588234753, −4.83820920366135048684888351016, −4.24639150197373705617632048545, −2.75958526171787397157693518539, −1.99129334535582377614296581428, 0, 1.99129334535582377614296581428, 2.75958526171787397157693518539, 4.24639150197373705617632048545, 4.83820920366135048684888351016, 5.77117588712804088861588234753, 6.40684021302317291689859230052, 7.68838829936442250962379923547, 8.540145264607385820885226966358, 9.061811684785377215962391006203

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