Properties

Label 2-1441-1.1-c1-0-48
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  − 2.66·2-s + 0.662·3-s + 5.12·4-s − 2.84·5-s − 1.76·6-s − 0.180·7-s − 8.33·8-s − 2.56·9-s + 7.58·10-s − 11-s + 3.39·12-s + 1.26·13-s + 0.480·14-s − 1.88·15-s + 12.0·16-s + 1.06·17-s + 6.83·18-s + 4.26·19-s − 14.5·20-s − 0.119·21-s + 2.66·22-s + 9.55·23-s − 5.52·24-s + 3.08·25-s − 3.37·26-s − 3.68·27-s − 0.923·28-s + ⋯
L(s)  = 1  − 1.88·2-s + 0.382·3-s + 2.56·4-s − 1.27·5-s − 0.721·6-s − 0.0681·7-s − 2.94·8-s − 0.853·9-s + 2.40·10-s − 0.301·11-s + 0.979·12-s + 0.351·13-s + 0.128·14-s − 0.486·15-s + 3.00·16-s + 0.259·17-s + 1.61·18-s + 0.977·19-s − 3.25·20-s − 0.0260·21-s + 0.569·22-s + 1.99·23-s − 1.12·24-s + 0.617·25-s − 0.662·26-s − 0.709·27-s − 0.174·28-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 \( 1 + 2.66T + 2T^{2} \)
3 \( 1 - 0.662T + 3T^{2} \)
5 \( 1 + 2.84T + 5T^{2} \)
7 \( 1 + 0.180T + 7T^{2} \)
13 \( 1 - 1.26T + 13T^{2} \)
17 \( 1 - 1.06T + 17T^{2} \)
19 \( 1 - 4.26T + 19T^{2} \)
23 \( 1 - 9.55T + 23T^{2} \)
29 \( 1 - 2.29T + 29T^{2} \)
31 \( 1 - 5.97T + 31T^{2} \)
37 \( 1 + 9.40T + 37T^{2} \)
41 \( 1 + 9.00T + 41T^{2} \)
43 \( 1 - 8.24T + 43T^{2} \)
47 \( 1 - 0.606T + 47T^{2} \)
53 \( 1 + 11.1T + 53T^{2} \)
59 \( 1 + 10.4T + 59T^{2} \)
61 \( 1 - 3.71T + 61T^{2} \)
67 \( 1 + 2.18T + 67T^{2} \)
71 \( 1 + 9.13T + 71T^{2} \)
73 \( 1 - 0.113T + 73T^{2} \)
79 \( 1 + 9.53T + 79T^{2} \)
83 \( 1 - 7.25T + 83T^{2} \)
89 \( 1 + 7.31T + 89T^{2} \)
97 \( 1 - 0.678T + 97T^{2} \)
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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

−8.884783363132954108767100425587, −8.463430198514850599112618620983, −7.74130591405612970802529018928, −7.20939359193917785055077572603, −6.28900089714487399620361546844, −5.03960686063303205098034728777, −3.33538117513081054483150762461, −2.87258691131999248791011811911, −1.26672769494788302003924607420, 0, 1.26672769494788302003924607420, 2.87258691131999248791011811911, 3.33538117513081054483150762461, 5.03960686063303205098034728777, 6.28900089714487399620361546844, 7.20939359193917785055077572603, 7.74130591405612970802529018928, 8.463430198514850599112618620983, 8.884783363132954108767100425587

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