Properties

Label 2-9702-1.1-c1-0-57
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 − 3·5-s + 8-s − 3·10-s + 11-s + 6·13-s + 16-s + 5·17-s + 6·19-s − 3·20-s + 22-s − 5·23-s + 4·25-s + 6·26-s + 6·29-s + 4·31-s + 32-s + 5·34-s − 2·37-s + 6·38-s − 3·40-s − 5·41-s − 10·43-s + 44-s − 5·46-s − 9·47-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 1.34·5-s + 0.353·8-s − 0.948·10-s + 0.301·11-s + 1.66·13-s + 1/4·16-s + 1.21·17-s + 1.37·19-s − 0.670·20-s + 0.213·22-s − 1.04·23-s + 4/5·25-s + 1.17·26-s + 1.11·29-s + 0.718·31-s + 0.176·32-s + 0.857·34-s − 0.328·37-s + 0.973·38-s − 0.474·40-s − 0.780·41-s − 1.52·43-s + 0.150·44-s − 0.737·46-s − 1.31·47-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: $\chi_{9702} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9702,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.110004611\)
\(L(\frac12)\) \(\approx\) \(3.110004611\)
\(L(\frac{3}{2})\) 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 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 9 T + p T^{2} \)
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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

−7.78985164827121356958698172807, −6.86969579364814092774886375109, −6.36432859325799059918110336209, −5.54423582813924940476279682580, −4.86993200973878544674445894406, −4.03164938769803071734816121409, −3.44808584822434878005567436131, −3.13351431247508337237661039953, −1.63656291905460921682623916998, −0.803813792845549000530706136404, 0.803813792845549000530706136404, 1.63656291905460921682623916998, 3.13351431247508337237661039953, 3.44808584822434878005567436131, 4.03164938769803071734816121409, 4.86993200973878544674445894406, 5.54423582813924940476279682580, 6.36432859325799059918110336209, 6.86969579364814092774886375109, 7.78985164827121356958698172807

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