Properties

Label 2-66654-1.1-c1-0-24
Degree $2$
Conductor $66654$
Sign $1$
Analytic cond. $532.234$
Root an. cond. $23.0702$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 7-s + 8-s + 4·11-s − 14-s + 16-s + 6·17-s + 6·19-s + 4·22-s − 5·25-s − 28-s − 10·29-s + 4·31-s + 32-s + 6·34-s + 2·37-s + 6·38-s + 10·41-s + 4·43-s + 4·44-s − 12·47-s + 49-s − 5·50-s − 6·53-s − 56-s − 10·58-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s + 1.20·11-s − 0.267·14-s + 1/4·16-s + 1.45·17-s + 1.37·19-s + 0.852·22-s − 25-s − 0.188·28-s − 1.85·29-s + 0.718·31-s + 0.176·32-s + 1.02·34-s + 0.328·37-s + 0.973·38-s + 1.56·41-s + 0.609·43-s + 0.603·44-s − 1.75·47-s + 1/7·49-s − 0.707·50-s − 0.824·53-s − 0.133·56-s − 1.31·58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 66654 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66654 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(66654\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(532.234\)
Root analytic conductor: \(23.0702\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{66654} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 66654,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.922119754\)
\(L(\frac12)\) \(\approx\) \(4.922119754\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
7 \( 1 + T \)
23 \( 1 \)
good5 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−14.16630693058600, −13.89287250930063, −13.04338640670130, −12.83955270482338, −12.12405337655661, −11.77680955922478, −11.32674785647722, −10.89019190680716, −9.876561937850978, −9.701594262131962, −9.358160726802383, −8.493848657513707, −7.763406982273617, −7.488569814428203, −6.889027052728525, −6.160550603266127, −5.781846148282814, −5.360918488507144, −4.509677399679625, −3.990897492296087, −3.351754002053125, −3.083218449226977, −2.049374877074225, −1.406293551129801, −0.6915699768351328, 0.6915699768351328, 1.406293551129801, 2.049374877074225, 3.083218449226977, 3.351754002053125, 3.990897492296087, 4.509677399679625, 5.360918488507144, 5.781846148282814, 6.160550603266127, 6.889027052728525, 7.488569814428203, 7.763406982273617, 8.493848657513707, 9.358160726802383, 9.701594262131962, 9.876561937850978, 10.89019190680716, 11.32674785647722, 11.77680955922478, 12.12405337655661, 12.83955270482338, 13.04338640670130, 13.89287250930063, 14.16630693058600

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