Properties

Label 2-532-1.1-c1-0-1
Degree $2$
Conductor $532$
Sign $1$
Analytic cond. $4.24804$
Root an. cond. $2.06107$
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  − 2.79·3-s + 3·5-s + 7-s + 4.79·9-s − 3.79·11-s − 13-s − 8.37·15-s + 3.79·17-s + 19-s − 2.79·21-s + 4.58·23-s + 4·25-s − 4.99·27-s + 3.79·29-s + 7.37·31-s + 10.5·33-s + 3·35-s + 5·37-s + 2.79·39-s − 3.79·41-s + 2·43-s + 14.3·45-s + 10.5·47-s + 49-s − 10.5·51-s − 8.37·53-s − 11.3·55-s + ⋯
L(s)  = 1  − 1.61·3-s + 1.34·5-s + 0.377·7-s + 1.59·9-s − 1.14·11-s − 0.277·13-s − 2.16·15-s + 0.919·17-s + 0.229·19-s − 0.609·21-s + 0.955·23-s + 0.800·25-s − 0.962·27-s + 0.704·29-s + 1.32·31-s + 1.84·33-s + 0.507·35-s + 0.821·37-s + 0.446·39-s − 0.592·41-s + 0.304·43-s + 2.14·45-s + 1.54·47-s + 0.142·49-s − 1.48·51-s − 1.15·53-s − 1.53·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(532\)    =    \(2^{2} \cdot 7 \cdot 19\)
Sign: $1$
Analytic conductor: \(4.24804\)
Root analytic conductor: \(2.06107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 532,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.090349859\)
\(L(\frac12)\) \(\approx\) \(1.090349859\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - T \)
19 \( 1 - T \)
good3 \( 1 + 2.79T + 3T^{2} \)
5 \( 1 - 3T + 5T^{2} \)
11 \( 1 + 3.79T + 11T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 - 3.79T + 17T^{2} \)
23 \( 1 - 4.58T + 23T^{2} \)
29 \( 1 - 3.79T + 29T^{2} \)
31 \( 1 - 7.37T + 31T^{2} \)
37 \( 1 - 5T + 37T^{2} \)
41 \( 1 + 3.79T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 - 10.5T + 47T^{2} \)
53 \( 1 + 8.37T + 53T^{2} \)
59 \( 1 - 12.1T + 59T^{2} \)
61 \( 1 + T + 61T^{2} \)
67 \( 1 + 9.37T + 67T^{2} \)
71 \( 1 + 12.1T + 71T^{2} \)
73 \( 1 - 16.3T + 73T^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 - 14.3T + 83T^{2} \)
89 \( 1 - 7.58T + 89T^{2} \)
97 \( 1 + 7T + 97T^{2} \)
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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

−10.64758075775048259794936033763, −10.27252764473795337469734156708, −9.427178393871595796018542308753, −8.025902644879022791631402380467, −6.93206250461869467950400949115, −5.98232778663916501699604567658, −5.37300311549347629519308899045, −4.70845617294110711937170327661, −2.64284162634064698467222211686, −1.07997414835013747402324904307, 1.07997414835013747402324904307, 2.64284162634064698467222211686, 4.70845617294110711937170327661, 5.37300311549347629519308899045, 5.98232778663916501699604567658, 6.93206250461869467950400949115, 8.025902644879022791631402380467, 9.427178393871595796018542308753, 10.27252764473795337469734156708, 10.64758075775048259794936033763

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