Properties

Label 2-189618-1.1-c1-0-28
Degree $2$
Conductor $189618$
Sign $-1$
Analytic cond. $1514.10$
Root an. cond. $38.9115$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 2·5-s − 6-s + 2·7-s + 8-s + 9-s − 2·10-s + 11-s − 12-s + 2·14-s + 2·15-s + 16-s − 17-s + 18-s + 2·19-s − 2·20-s − 2·21-s + 22-s − 6·23-s − 24-s − 25-s − 27-s + 2·28-s + 2·29-s + 2·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.301·11-s − 0.288·12-s + 0.534·14-s + 0.516·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 0.458·19-s − 0.447·20-s − 0.436·21-s + 0.213·22-s − 1.25·23-s − 0.204·24-s − 1/5·25-s − 0.192·27-s + 0.377·28-s + 0.371·29-s + 0.365·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189618\)    =    \(2 \cdot 3 \cdot 11 \cdot 13^{2} \cdot 17\)
Sign: $-1$
Analytic conductor: \(1514.10\)
Root analytic conductor: \(38.9115\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 189618,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 \)
17 \( 1 + T \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−13.36617398236076, −12.78828160085700, −12.24477130207090, −11.87319077946192, −11.62306866002762, −11.24460383099493, −10.64722325683169, −10.24774482803839, −9.667559581231790, −9.099559322590109, −8.299349865521237, −8.027548597717235, −7.658358758238872, −6.900604985153844, −6.641364867049570, −5.984253214657607, −5.478999822069936, −4.926162667947444, −4.495492324621473, −4.064048005378783, −3.457838585635134, −3.018220278252665, −1.929475407480767, −1.741005864159193, −0.7794103014452556, 0, 0.7794103014452556, 1.741005864159193, 1.929475407480767, 3.018220278252665, 3.457838585635134, 4.064048005378783, 4.495492324621473, 4.926162667947444, 5.478999822069936, 5.984253214657607, 6.641364867049570, 6.900604985153844, 7.658358758238872, 8.027548597717235, 8.299349865521237, 9.099559322590109, 9.667559581231790, 10.24774482803839, 10.64722325683169, 11.24460383099493, 11.62306866002762, 11.87319077946192, 12.24477130207090, 12.78828160085700, 13.36617398236076

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