Properties

Label 2-1441-1.1-c1-0-47
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  − 0.187·2-s + 3.03·3-s − 1.96·4-s + 3.17·5-s − 0.568·6-s − 3.01·7-s + 0.742·8-s + 6.21·9-s − 0.593·10-s − 11-s − 5.96·12-s − 3.91·13-s + 0.563·14-s + 9.62·15-s + 3.79·16-s + 5.98·17-s − 1.16·18-s + 1.02·19-s − 6.23·20-s − 9.13·21-s + 0.187·22-s + 3.44·23-s + 2.25·24-s + 5.06·25-s + 0.733·26-s + 9.74·27-s + 5.91·28-s + ⋯
L(s)  = 1  − 0.132·2-s + 1.75·3-s − 0.982·4-s + 1.41·5-s − 0.231·6-s − 1.13·7-s + 0.262·8-s + 2.07·9-s − 0.187·10-s − 0.301·11-s − 1.72·12-s − 1.08·13-s + 0.150·14-s + 2.48·15-s + 0.947·16-s + 1.45·17-s − 0.274·18-s + 0.234·19-s − 1.39·20-s − 1.99·21-s + 0.0399·22-s + 0.718·23-s + 0.459·24-s + 1.01·25-s + 0.143·26-s + 1.87·27-s + 1.11·28-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: \(0\)
Selberg data: \((2,\ 1441,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.772484211\)
\(L(\frac12)\) \(\approx\) \(2.772484211\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.187T + 2T^{2} \)
3 \( 1 - 3.03T + 3T^{2} \)
5 \( 1 - 3.17T + 5T^{2} \)
7 \( 1 + 3.01T + 7T^{2} \)
13 \( 1 + 3.91T + 13T^{2} \)
17 \( 1 - 5.98T + 17T^{2} \)
19 \( 1 - 1.02T + 19T^{2} \)
23 \( 1 - 3.44T + 23T^{2} \)
29 \( 1 - 6.73T + 29T^{2} \)
31 \( 1 - 7.30T + 31T^{2} \)
37 \( 1 - 4.80T + 37T^{2} \)
41 \( 1 + 4.42T + 41T^{2} \)
43 \( 1 - 5.88T + 43T^{2} \)
47 \( 1 - 8.97T + 47T^{2} \)
53 \( 1 + 9.57T + 53T^{2} \)
59 \( 1 + 6.67T + 59T^{2} \)
61 \( 1 + 6.28T + 61T^{2} \)
67 \( 1 + 2.79T + 67T^{2} \)
71 \( 1 - 4.10T + 71T^{2} \)
73 \( 1 - 2.85T + 73T^{2} \)
79 \( 1 + 12.0T + 79T^{2} \)
83 \( 1 + 8.30T + 83T^{2} \)
89 \( 1 + 17.2T + 89T^{2} \)
97 \( 1 - 0.194T + 97T^{2} \)
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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

−9.512523708735240307615422433749, −8.985693622579615433780849507299, −8.085072850878738980206064244578, −7.38195258834535601053680206079, −6.30034530107519817904391134433, −5.28464849258543149461501766578, −4.32696513023353416899227497151, −3.03181039680523042472488351285, −2.74478924692601768543271323202, −1.25729023186846704835574786047, 1.25729023186846704835574786047, 2.74478924692601768543271323202, 3.03181039680523042472488351285, 4.32696513023353416899227497151, 5.28464849258543149461501766578, 6.30034530107519817904391134433, 7.38195258834535601053680206079, 8.085072850878738980206064244578, 8.985693622579615433780849507299, 9.512523708735240307615422433749

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