Properties

Label 2-9702-1.1-c1-0-109
Degree $2$
Conductor $9702$
Sign $-1$
Analytic cond. $77.4708$
Root an. cond. $8.80175$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 8-s + 11-s − 2·13-s + 16-s − 6·17-s − 2·19-s − 22-s − 5·25-s + 2·26-s + 6·29-s + 4·31-s − 32-s + 6·34-s + 2·37-s + 2·38-s + 6·41-s − 10·43-s + 44-s + 12·47-s + 5·50-s − 2·52-s + 12·53-s − 6·58-s + 12·59-s + 10·61-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s + 0.301·11-s − 0.554·13-s + 1/4·16-s − 1.45·17-s − 0.458·19-s − 0.213·22-s − 25-s + 0.392·26-s + 1.11·29-s + 0.718·31-s − 0.176·32-s + 1.02·34-s + 0.328·37-s + 0.324·38-s + 0.937·41-s − 1.52·43-s + 0.150·44-s + 1.75·47-s + 0.707·50-s − 0.277·52-s + 1.64·53-s − 0.787·58-s + 1.56·59-s + 1.28·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9702\)    =    \(2 \cdot 3^{2} \cdot 7^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(77.4708\)
Root analytic conductor: \(8.80175\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 9702,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 \)
11 \( 1 - T \)
good5 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.22256681200124913942068687753, −6.86319691613869780218188863622, −6.13488512587456810544110169825, −5.40474200258705758947525517831, −4.41604514672277597522072986408, −3.93582844976437782803839905400, −2.64565898313481655439998497825, −2.25338768348159295239588485846, −1.09696957382779221676747338783, 0, 1.09696957382779221676747338783, 2.25338768348159295239588485846, 2.64565898313481655439998497825, 3.93582844976437782803839905400, 4.41604514672277597522072986408, 5.40474200258705758947525517831, 6.13488512587456810544110169825, 6.86319691613869780218188863622, 7.22256681200124913942068687753

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