Properties

Label 2-189618-1.1-c1-0-18
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 6-s + 4·7-s + 8-s + 9-s − 11-s + 12-s + 4·14-s + 16-s + 17-s + 18-s − 2·19-s + 4·21-s − 22-s + 24-s − 5·25-s + 27-s + 4·28-s − 6·29-s − 8·31-s + 32-s − 33-s + 34-s + 36-s − 2·37-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s + 1.06·14-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 0.458·19-s + 0.872·21-s − 0.213·22-s + 0.204·24-s − 25-s + 0.192·27-s + 0.755·28-s − 1.11·29-s − 1.43·31-s + 0.176·32-s − 0.174·33-s + 0.171·34-s + 1/6·36-s − 0.328·37-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: $\chi_{189618} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 189618,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.712983738\)
\(L(\frac12)\) \(\approx\) \(6.712983738\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−13.08142626709802, −12.77578753167145, −12.18477990243577, −11.80403846408051, −11.12188994868082, −10.94086836490262, −10.50992187736094, −9.751407575471143, −9.319507084126064, −8.736395577506440, −8.259676665957555, −7.769529193229499, −7.378421346652463, −7.075561918378156, −6.083802212588082, −5.754269734718197, −5.213038085257280, −4.740638070291137, −4.111947321591945, −3.796103214169029, −3.161256344316039, −2.265499236703333, −2.057644695847902, −1.488525943375051, −0.5950829907749451, 0.5950829907749451, 1.488525943375051, 2.057644695847902, 2.265499236703333, 3.161256344316039, 3.796103214169029, 4.111947321591945, 4.740638070291137, 5.213038085257280, 5.754269734718197, 6.083802212588082, 7.075561918378156, 7.378421346652463, 7.769529193229499, 8.259676665957555, 8.736395577506440, 9.319507084126064, 9.751407575471143, 10.50992187736094, 10.94086836490262, 11.12188994868082, 11.80403846408051, 12.18477990243577, 12.77578753167145, 13.08142626709802

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