Properties

Label 2-206310-1.1-c1-0-3
Degree $2$
Conductor $206310$
Sign $1$
Analytic cond. $1647.39$
Root an. cond. $40.5880$
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 − 3-s + 4-s + 5-s − 6-s − 2·7-s + 8-s + 9-s + 10-s − 4·11-s − 12-s − 13-s − 2·14-s − 15-s + 16-s − 8·17-s + 18-s + 6·19-s + 20-s + 2·21-s − 4·22-s − 24-s + 25-s − 26-s − 27-s − 2·28-s − 4·29-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s − 0.288·12-s − 0.277·13-s − 0.534·14-s − 0.258·15-s + 1/4·16-s − 1.94·17-s + 0.235·18-s + 1.37·19-s + 0.223·20-s + 0.436·21-s − 0.852·22-s − 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s − 0.377·28-s − 0.742·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 206310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 206310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(206310\)    =    \(2 \cdot 3 \cdot 5 \cdot 13 \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(1647.39\)
Root analytic conductor: \(40.5880\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{206310} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 206310,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.101216782\)
\(L(\frac12)\) \(\approx\) \(1.101216782\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
5 \( 1 - T \)
13 \( 1 + T \)
23 \( 1 \)
good7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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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

−13.17686104114199, −12.78619672807010, −12.04688869363038, −11.71181997760139, −11.24795743043209, −10.68890655055334, −10.35934931647443, −9.843143767217985, −9.382185682654074, −8.855998836803155, −8.270998338097052, −7.458953886148569, −7.234001895569442, −6.738075648495513, −6.136993552363498, −5.733505985202792, −5.277248299723385, −4.826314001625720, −4.247369123273052, −3.707487032247443, −2.904427654258655, −2.599113125790174, −1.996565801258197, −1.206631329026830, −0.2674366864846795, 0.2674366864846795, 1.206631329026830, 1.996565801258197, 2.599113125790174, 2.904427654258655, 3.707487032247443, 4.247369123273052, 4.826314001625720, 5.277248299723385, 5.733505985202792, 6.136993552363498, 6.738075648495513, 7.234001895569442, 7.458953886148569, 8.270998338097052, 8.855998836803155, 9.382185682654074, 9.843143767217985, 10.35934931647443, 10.68890655055334, 11.24795743043209, 11.71181997760139, 12.04688869363038, 12.78619672807010, 13.17686104114199

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