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Properties

Label 2-182070-1.1-c1-0-44
Degree $2$
Conductor $182070$
Sign $-1$
Analytic cond. $1453.83$
Root an. cond. $38.1292$
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 + 4-s + 5-s − 7-s − 8-s − 10-s − 4·11-s − 6·13-s + 14-s + 16-s + 20-s + 4·22-s + 4·23-s + 25-s + 6·26-s − 28-s − 6·29-s − 32-s − 35-s + 6·37-s − 40-s − 6·41-s − 4·43-s − 4·44-s − 4·46-s + 12·47-s + 49-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s − 0.353·8-s − 0.316·10-s − 1.20·11-s − 1.66·13-s + 0.267·14-s + 1/4·16-s + 0.223·20-s + 0.852·22-s + 0.834·23-s + 1/5·25-s + 1.17·26-s − 0.188·28-s − 1.11·29-s − 0.176·32-s − 0.169·35-s + 0.986·37-s − 0.158·40-s − 0.937·41-s − 0.609·43-s − 0.603·44-s − 0.589·46-s + 1.75·47-s + 1/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 182070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(182070\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(1453.83\)
Root analytic conductor: \(38.1292\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 182070,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 - T \)
7 \( 1 + T \)
17 \( 1 \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−13.21460040188740, −12.96852843161311, −12.43161487133862, −11.98696521819353, −11.45755849960740, −10.77993947503828, −10.58934556733102, −9.889425579969416, −9.709133848932782, −9.233930969260258, −8.662469130267348, −8.129600681983185, −7.501286965488791, −7.297593961346966, −6.788183765800864, −6.098831290237901, −5.571747908478230, −5.117696052046339, −4.670775229543162, −3.865569065901870, −3.081932786372119, −2.644758609492152, −2.234748259145222, −1.544258566322888, −0.6192337485826385, 0, 0.6192337485826385, 1.544258566322888, 2.234748259145222, 2.644758609492152, 3.081932786372119, 3.865569065901870, 4.670775229543162, 5.117696052046339, 5.571747908478230, 6.098831290237901, 6.788183765800864, 7.297593961346966, 7.501286965488791, 8.129600681983185, 8.662469130267348, 9.233930969260258, 9.709133848932782, 9.889425579969416, 10.58934556733102, 10.77993947503828, 11.45755849960740, 11.98696521819353, 12.43161487133862, 12.96852843161311, 13.21460040188740

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