Properties

Label 2-39e2-1.1-c3-0-30
Degree $2$
Conductor $1521$
Sign $1$
Analytic cond. $89.7419$
Root an. cond. $9.47322$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.17·2-s + 2.07·4-s − 6.74·5-s − 14.1·7-s − 18.8·8-s − 21.3·10-s − 62.4·11-s − 44.9·14-s − 76.2·16-s + 58.6·17-s + 64.1·19-s − 13.9·20-s − 198.·22-s − 10.9·23-s − 79.5·25-s − 29.3·28-s − 216.·29-s − 38.6·31-s − 91.5·32-s + 186.·34-s + 95.5·35-s + 423.·37-s + 203.·38-s + 126.·40-s + 366.·41-s − 128.·43-s − 129.·44-s + ⋯
L(s)  = 1  + 1.12·2-s + 0.258·4-s − 0.602·5-s − 0.765·7-s − 0.831·8-s − 0.676·10-s − 1.71·11-s − 0.858·14-s − 1.19·16-s + 0.836·17-s + 0.774·19-s − 0.156·20-s − 1.92·22-s − 0.0990·23-s − 0.636·25-s − 0.198·28-s − 1.38·29-s − 0.223·31-s − 0.505·32-s + 0.938·34-s + 0.461·35-s + 1.88·37-s + 0.869·38-s + 0.501·40-s + 1.39·41-s − 0.455·43-s − 0.443·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(89.7419\)
Root analytic conductor: \(9.47322\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1521,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.496190744\)
\(L(\frac12)\) \(\approx\) \(1.496190744\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 - 3.17T + 8T^{2} \)
5 \( 1 + 6.74T + 125T^{2} \)
7 \( 1 + 14.1T + 343T^{2} \)
11 \( 1 + 62.4T + 1.33e3T^{2} \)
17 \( 1 - 58.6T + 4.91e3T^{2} \)
19 \( 1 - 64.1T + 6.85e3T^{2} \)
23 \( 1 + 10.9T + 1.21e4T^{2} \)
29 \( 1 + 216.T + 2.43e4T^{2} \)
31 \( 1 + 38.6T + 2.97e4T^{2} \)
37 \( 1 - 423.T + 5.06e4T^{2} \)
41 \( 1 - 366.T + 6.89e4T^{2} \)
43 \( 1 + 128.T + 7.95e4T^{2} \)
47 \( 1 - 93.1T + 1.03e5T^{2} \)
53 \( 1 + 131.T + 1.48e5T^{2} \)
59 \( 1 + 386.T + 2.05e5T^{2} \)
61 \( 1 + 621.T + 2.26e5T^{2} \)
67 \( 1 - 865.T + 3.00e5T^{2} \)
71 \( 1 - 607.T + 3.57e5T^{2} \)
73 \( 1 - 980.T + 3.89e5T^{2} \)
79 \( 1 - 1.33e3T + 4.93e5T^{2} \)
83 \( 1 + 907.T + 5.71e5T^{2} \)
89 \( 1 - 1.03e3T + 7.04e5T^{2} \)
97 \( 1 - 1.04e3T + 9.12e5T^{2} \)
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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

−9.333060736249340697053166230592, −7.86803806342114058061372514246, −7.70100743409639508573327481737, −6.37015536736147715622613786402, −5.61255997792977407906769926749, −5.00300040230651945125646844415, −3.93683702909906053686962607618, −3.26254144575299950498854489098, −2.44524840430735815530352293612, −0.47455874685598872431166464370, 0.47455874685598872431166464370, 2.44524840430735815530352293612, 3.26254144575299950498854489098, 3.93683702909906053686962607618, 5.00300040230651945125646844415, 5.61255997792977407906769926749, 6.37015536736147715622613786402, 7.70100743409639508573327481737, 7.86803806342114058061372514246, 9.333060736249340697053166230592

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