Properties

Label 2-1441-1.1-c1-0-5
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  + 0.800·2-s − 3.25·3-s − 1.35·4-s + 0.350·5-s − 2.60·6-s − 2.72·7-s − 2.68·8-s + 7.61·9-s + 0.280·10-s − 11-s + 4.42·12-s − 6.49·13-s − 2.18·14-s − 1.14·15-s + 0.566·16-s − 7.27·17-s + 6.09·18-s + 2.26·19-s − 0.475·20-s + 8.88·21-s − 0.800·22-s − 0.681·23-s + 8.76·24-s − 4.87·25-s − 5.20·26-s − 15.0·27-s + 3.70·28-s + ⋯
L(s)  = 1  + 0.565·2-s − 1.88·3-s − 0.679·4-s + 0.156·5-s − 1.06·6-s − 1.03·7-s − 0.950·8-s + 2.53·9-s + 0.0886·10-s − 0.301·11-s + 1.27·12-s − 1.80·13-s − 0.583·14-s − 0.294·15-s + 0.141·16-s − 1.76·17-s + 1.43·18-s + 0.519·19-s − 0.106·20-s + 1.93·21-s − 0.170·22-s − 0.142·23-s + 1.78·24-s − 0.975·25-s − 1.01·26-s − 2.89·27-s + 0.700·28-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

\(L(1)\) \(\approx\) \(0.2263833762\)
\(L(\frac12)\) \(\approx\) \(0.2263833762\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.800T + 2T^{2} \)
3 \( 1 + 3.25T + 3T^{2} \)
5 \( 1 - 0.350T + 5T^{2} \)
7 \( 1 + 2.72T + 7T^{2} \)
13 \( 1 + 6.49T + 13T^{2} \)
17 \( 1 + 7.27T + 17T^{2} \)
19 \( 1 - 2.26T + 19T^{2} \)
23 \( 1 + 0.681T + 23T^{2} \)
29 \( 1 + 8.89T + 29T^{2} \)
31 \( 1 - 1.04T + 31T^{2} \)
37 \( 1 + 4.67T + 37T^{2} \)
41 \( 1 - 7.70T + 41T^{2} \)
43 \( 1 - 7.15T + 43T^{2} \)
47 \( 1 - 10.9T + 47T^{2} \)
53 \( 1 - 10.3T + 53T^{2} \)
59 \( 1 - 1.67T + 59T^{2} \)
61 \( 1 + 4.14T + 61T^{2} \)
67 \( 1 - 7.24T + 67T^{2} \)
71 \( 1 + 1.59T + 71T^{2} \)
73 \( 1 + 11.8T + 73T^{2} \)
79 \( 1 - 2.14T + 79T^{2} \)
83 \( 1 + 13.4T + 83T^{2} \)
89 \( 1 + 2.83T + 89T^{2} \)
97 \( 1 - 16.4T + 97T^{2} \)
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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

−9.648936060042106159331557183297, −9.149175056592033511556315823906, −7.45774261099302842359820089190, −6.89849836011975874102379779532, −5.86114512810405306751175453751, −5.53383094329133385890132598010, −4.58107999671171901280954428587, −3.98361734286844767755260308760, −2.39214624586894634624020952770, −0.32362616428610088763762879238, 0.32362616428610088763762879238, 2.39214624586894634624020952770, 3.98361734286844767755260308760, 4.58107999671171901280954428587, 5.53383094329133385890132598010, 5.86114512810405306751175453751, 6.89849836011975874102379779532, 7.45774261099302842359820089190, 9.149175056592033511556315823906, 9.648936060042106159331557183297

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