Properties

Label 2-1441-1.1-c1-0-12
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.07·2-s − 2.88·3-s + 2.30·4-s − 3.62·5-s − 5.97·6-s − 5.05·7-s + 0.639·8-s + 5.29·9-s − 7.52·10-s + 11-s − 6.64·12-s + 0.599·13-s − 10.5·14-s + 10.4·15-s − 3.28·16-s − 3.13·17-s + 10.9·18-s + 3.58·19-s − 8.37·20-s + 14.5·21-s + 2.07·22-s + 6.21·23-s − 1.84·24-s + 8.15·25-s + 1.24·26-s − 6.61·27-s − 11.6·28-s + ⋯
L(s)  = 1  + 1.46·2-s − 1.66·3-s + 1.15·4-s − 1.62·5-s − 2.44·6-s − 1.91·7-s + 0.225·8-s + 1.76·9-s − 2.38·10-s + 0.301·11-s − 1.91·12-s + 0.166·13-s − 2.80·14-s + 2.69·15-s − 0.822·16-s − 0.760·17-s + 2.59·18-s + 0.821·19-s − 1.87·20-s + 3.18·21-s + 0.442·22-s + 1.29·23-s − 0.375·24-s + 1.63·25-s + 0.243·26-s − 1.27·27-s − 2.20·28-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: \(0\)
Selberg data: \((2,\ 1441,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7265063158\)
\(L(\frac12)\) \(\approx\) \(0.7265063158\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.07T + 2T^{2} \)
3 \( 1 + 2.88T + 3T^{2} \)
5 \( 1 + 3.62T + 5T^{2} \)
7 \( 1 + 5.05T + 7T^{2} \)
13 \( 1 - 0.599T + 13T^{2} \)
17 \( 1 + 3.13T + 17T^{2} \)
19 \( 1 - 3.58T + 19T^{2} \)
23 \( 1 - 6.21T + 23T^{2} \)
29 \( 1 - 1.58T + 29T^{2} \)
31 \( 1 + 1.68T + 31T^{2} \)
37 \( 1 - 0.605T + 37T^{2} \)
41 \( 1 + 10.4T + 41T^{2} \)
43 \( 1 - 0.0776T + 43T^{2} \)
47 \( 1 + 2.17T + 47T^{2} \)
53 \( 1 + 0.650T + 53T^{2} \)
59 \( 1 - 12.5T + 59T^{2} \)
61 \( 1 + 10.8T + 61T^{2} \)
67 \( 1 + 13.6T + 67T^{2} \)
71 \( 1 - 15.3T + 71T^{2} \)
73 \( 1 - 9.83T + 73T^{2} \)
79 \( 1 - 1.02T + 79T^{2} \)
83 \( 1 + 3.03T + 83T^{2} \)
89 \( 1 + 7.74T + 89T^{2} \)
97 \( 1 + 7.34T + 97T^{2} \)
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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

−9.756737454725670930731950858689, −8.777334119933410008254460872511, −7.18902485418242099445529749770, −6.81044760607319339823106404321, −6.19039953407400971171171355382, −5.26482829677090974696148078234, −4.52288351786580693386980330521, −3.67848233104027446058526348078, −3.09859287669499547292564674512, −0.50718316784784106930791764868, 0.50718316784784106930791764868, 3.09859287669499547292564674512, 3.67848233104027446058526348078, 4.52288351786580693386980330521, 5.26482829677090974696148078234, 6.19039953407400971171171355382, 6.81044760607319339823106404321, 7.18902485418242099445529749770, 8.777334119933410008254460872511, 9.756737454725670930731950858689

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