Properties

Label 2-189618-1.1-c1-0-13
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 + 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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 189618,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.375447918\)
\(L(\frac12)\) \(\approx\) \(5.375447918\)
\(L(\frac{3}{2})\) 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

−13.37759890120240, −12.59592546870651, −12.17883848959490, −11.85199262336367, −11.33932014336061, −10.81349616092228, −10.28497809344241, −9.850826811421163, −9.450451370659498, −8.624396908759015, −8.265077803776461, −7.846747125237137, −7.441098418513046, −6.858630051547696, −6.187209561289655, −5.795372505045112, −5.223588389888654, −4.679820350419139, −4.177384111819943, −3.674057394857314, −3.200325396932583, −2.341573474275397, −2.057395836822474, −1.458987240462181, −0.5297389760576355, 0.5297389760576355, 1.458987240462181, 2.057395836822474, 2.341573474275397, 3.200325396932583, 3.674057394857314, 4.177384111819943, 4.679820350419139, 5.223588389888654, 5.795372505045112, 6.187209561289655, 6.858630051547696, 7.441098418513046, 7.846747125237137, 8.265077803776461, 8.624396908759015, 9.450451370659498, 9.850826811421163, 10.28497809344241, 10.81349616092228, 11.33932014336061, 11.85199262336367, 12.17883848959490, 12.59592546870651, 13.37759890120240

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