Properties

Label 2-189618-1.1-c1-0-42
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 $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 6-s − 2·7-s − 8-s + 9-s + 11-s + 12-s + 2·14-s + 16-s + 17-s − 18-s − 2·21-s − 22-s − 8·23-s − 24-s − 5·25-s + 27-s − 2·28-s − 2·29-s − 4·31-s − 32-s + 33-s − 34-s + 36-s + 37-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.301·11-s + 0.288·12-s + 0.534·14-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.436·21-s − 0.213·22-s − 1.66·23-s − 0.204·24-s − 25-s + 0.192·27-s − 0.377·28-s − 0.371·29-s − 0.718·31-s − 0.176·32-s + 0.174·33-s − 0.171·34-s + 1/6·36-s + 0.164·37-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: \(2\)
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 \( 1 + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 - 7 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 15 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 + 5 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
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

−13.47141164691350, −13.27824160043274, −12.49278902077716, −12.23565869428281, −11.63097654312988, −11.29655955403855, −10.55368276961277, −10.10788155694102, −9.709880070693608, −9.496783274163750, −8.713005906863179, −8.528089867501359, −7.806339202255956, −7.487848310194320, −7.014231729139483, −6.315511672795641, −5.961879954789752, −5.535553878219903, −4.525326089589621, −4.152926246111736, −3.412309684811600, −3.134794799385523, −2.379333246105477, −1.720070964871811, −1.367172770981595, 0, 0, 1.367172770981595, 1.720070964871811, 2.379333246105477, 3.134794799385523, 3.412309684811600, 4.152926246111736, 4.525326089589621, 5.535553878219903, 5.961879954789752, 6.315511672795641, 7.014231729139483, 7.487848310194320, 7.806339202255956, 8.528089867501359, 8.713005906863179, 9.496783274163750, 9.709880070693608, 10.10788155694102, 10.55368276961277, 11.29655955403855, 11.63097654312988, 12.23565869428281, 12.49278902077716, 13.27824160043274, 13.47141164691350

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