Properties

Label 2-9702-1.1-c1-0-21
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 8-s + 11-s − 2·13-s + 16-s − 2·19-s − 22-s − 5·25-s + 2·26-s + 6·29-s − 2·31-s − 32-s + 2·37-s + 2·38-s − 4·43-s + 44-s − 6·47-s + 5·50-s − 2·52-s + 6·53-s − 6·58-s − 2·61-s + 2·62-s + 64-s − 4·67-s + 12·71-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 − 0.458·19-s − 0.213·22-s − 25-s + 0.392·26-s + 1.11·29-s − 0.359·31-s − 0.176·32-s + 0.328·37-s + 0.324·38-s − 0.609·43-s + 0.150·44-s − 0.875·47-s + 0.707·50-s − 0.277·52-s + 0.824·53-s − 0.787·58-s − 0.256·61-s + 0.254·62-s + 1/8·64-s − 0.488·67-s + 1.42·71-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: \(0\)
Selberg data: \((2,\ 9702,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.146606821\)
\(L(\frac12)\) \(\approx\) \(1.146606821\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
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

−7.85753432302505928921103293660, −6.98338187441398347810360655747, −6.52121111519916052383579561807, −5.75791670905845557167098435741, −4.96022111668989750761323800786, −4.15948019503367063738020240961, −3.31912164781038622756964020494, −2.42588747445440968215441279751, −1.68581041078581093085939022337, −0.56363101267073554687128119537, 0.56363101267073554687128119537, 1.68581041078581093085939022337, 2.42588747445440968215441279751, 3.31912164781038622756964020494, 4.15948019503367063738020240961, 4.96022111668989750761323800786, 5.75791670905845557167098435741, 6.52121111519916052383579561807, 6.98338187441398347810360655747, 7.85753432302505928921103293660

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