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Properties

Label 2-189630-1.1-c1-0-22
Degree $2$
Conductor $189630$
Sign $1$
Analytic cond. $1514.20$
Root an. cond. $38.9127$
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 − 5-s + 8-s − 10-s + 2·13-s + 16-s + 2·17-s − 4·19-s − 20-s + 25-s + 2·26-s − 2·29-s + 32-s + 2·34-s + 10·37-s − 4·38-s − 40-s − 6·41-s − 43-s − 8·47-s + 50-s + 2·52-s − 2·53-s − 2·58-s − 2·61-s + 64-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s + 0.554·13-s + 1/4·16-s + 0.485·17-s − 0.917·19-s − 0.223·20-s + 1/5·25-s + 0.392·26-s − 0.371·29-s + 0.176·32-s + 0.342·34-s + 1.64·37-s − 0.648·38-s − 0.158·40-s − 0.937·41-s − 0.152·43-s − 1.16·47-s + 0.141·50-s + 0.277·52-s − 0.274·53-s − 0.262·58-s − 0.256·61-s + 1/8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 43\)
Sign: $1$
Analytic conductor: \(1514.20\)
Root analytic conductor: \(38.9127\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 189630,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.897090496\)
\(L(\frac12)\) \(\approx\) \(2.897090496\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
43 \( 1 + T \)
good11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−13.14773977633574, −12.77307803723328, −12.13705524849982, −11.76095524501335, −11.33254373732449, −10.89229281810841, −10.36039570405418, −9.969075383443634, −9.274552519520973, −8.800869171424970, −8.240921084177257, −7.775797424421435, −7.410877009638258, −6.613869819079557, −6.365029459989282, −5.826455952750434, −5.215191540875661, −4.705314554175594, −4.146423767973462, −3.766608021855977, −3.083992283644615, −2.690836817987768, −1.821609054661473, −1.345574124332930, −0.4219091961686063, 0.4219091961686063, 1.345574124332930, 1.821609054661473, 2.690836817987768, 3.083992283644615, 3.766608021855977, 4.146423767973462, 4.705314554175594, 5.215191540875661, 5.826455952750434, 6.365029459989282, 6.613869819079557, 7.410877009638258, 7.775797424421435, 8.240921084177257, 8.800869171424970, 9.274552519520973, 9.969075383443634, 10.36039570405418, 10.89229281810841, 11.33254373732449, 11.76095524501335, 12.13705524849982, 12.77307803723328, 13.14773977633574

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