Properties

Label 2-12138-1.1-c1-0-29
Degree $2$
Conductor $12138$
Sign $-1$
Analytic cond. $96.9224$
Root an. cond. $9.84491$
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 − 7-s + 8-s + 9-s + 2·10-s − 4·11-s + 12-s − 2·13-s − 14-s + 2·15-s + 16-s + 18-s + 4·19-s + 2·20-s − 21-s − 4·22-s − 8·23-s + 24-s − 25-s − 2·26-s + 27-s − 28-s − 6·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.632·10-s − 1.20·11-s + 0.288·12-s − 0.554·13-s − 0.267·14-s + 0.516·15-s + 1/4·16-s + 0.235·18-s + 0.917·19-s + 0.447·20-s − 0.218·21-s − 0.852·22-s − 1.66·23-s + 0.204·24-s − 1/5·25-s − 0.392·26-s + 0.192·27-s − 0.188·28-s − 1.11·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12138\)    =    \(2 \cdot 3 \cdot 7 \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(96.9224\)
Root analytic conductor: \(9.84491\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{12138} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 12138,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 + T \)
17 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−16.62048109561241, −15.88156430921023, −15.43514565895425, −14.95428348508706, −14.14241003979634, −13.81493397859807, −13.32636717100551, −12.91015807959009, −12.16352991029155, −11.70841581683022, −10.80215341640397, −10.07399157467546, −9.898289221739233, −9.223107131361027, −8.304421690524827, −7.725934908411055, −7.199770262085365, −6.376090039530376, −5.667880567335022, −5.299651836061111, −4.452422896258439, −3.607476335167343, −2.937332237282683, −2.230758269600829, −1.643368690332513, 0, 1.643368690332513, 2.230758269600829, 2.937332237282683, 3.607476335167343, 4.452422896258439, 5.299651836061111, 5.667880567335022, 6.376090039530376, 7.199770262085365, 7.725934908411055, 8.304421690524827, 9.223107131361027, 9.898289221739233, 10.07399157467546, 10.80215341640397, 11.70841581683022, 12.16352991029155, 12.91015807959009, 13.32636717100551, 13.81493397859807, 14.14241003979634, 14.95428348508706, 15.43514565895425, 15.88156430921023, 16.62048109561241

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