Properties

Label 2-189618-1.1-c1-0-21
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 − 5-s + 6-s − 3·7-s − 8-s + 9-s + 10-s + 11-s − 12-s + 3·14-s + 15-s + 16-s + 17-s − 18-s + 4·19-s − 20-s + 3·21-s − 22-s − 3·23-s + 24-s − 4·25-s − 27-s − 3·28-s + 7·29-s − 30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 1.13·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s − 0.288·12-s + 0.801·14-s + 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.917·19-s − 0.223·20-s + 0.654·21-s − 0.213·22-s − 0.625·23-s + 0.204·24-s − 4/5·25-s − 0.192·27-s − 0.566·28-s + 1.29·29-s − 0.182·30-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 \( 1 + T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−13.33260958661491, −12.64859407239661, −12.29096834136896, −11.93717897911209, −11.41551429670593, −11.10378375642761, −10.35735307197978, −9.944084874462316, −9.720714281370189, −9.237531874957887, −8.554808161975742, −8.055340268702380, −7.676430599233306, −7.018043652613862, −6.633280566036002, −6.196016395494266, −5.736820950263230, −5.082524505834666, −4.467487502635361, −3.823137754899048, −3.289725366512114, −2.856375197107975, −2.023030554271240, −1.304267467513087, −0.6383786065629037, 0, 0.6383786065629037, 1.304267467513087, 2.023030554271240, 2.856375197107975, 3.289725366512114, 3.823137754899048, 4.467487502635361, 5.082524505834666, 5.736820950263230, 6.196016395494266, 6.633280566036002, 7.018043652613862, 7.676430599233306, 8.055340268702380, 8.554808161975742, 9.237531874957887, 9.720714281370189, 9.944084874462316, 10.35735307197978, 11.10378375642761, 11.41551429670593, 11.93717897911209, 12.29096834136896, 12.64859407239661, 13.33260958661491

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