Properties

Label 2-9702-1.1-c1-0-159
Degree $2$
Conductor $9702$
Sign $-1$
Analytic cond. $77.4708$
Root an. cond. $8.80175$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 8-s + 11-s + 2.82·13-s + 16-s + 2.82·17-s − 5.65·19-s + 22-s − 8·23-s − 5·25-s + 2.82·26-s − 2·29-s − 8.48·31-s + 32-s + 2.82·34-s + 2·37-s − 5.65·38-s + 2.82·41-s − 4·43-s + 44-s − 8·46-s + 2.82·47-s − 5·50-s + 2.82·52-s − 14·53-s − 2·58-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 0.353·8-s + 0.301·11-s + 0.784·13-s + 0.250·16-s + 0.685·17-s − 1.29·19-s + 0.213·22-s − 1.66·23-s − 25-s + 0.554·26-s − 0.371·29-s − 1.52·31-s + 0.176·32-s + 0.485·34-s + 0.328·37-s − 0.917·38-s + 0.441·41-s − 0.609·43-s + 0.150·44-s − 1.17·46-s + 0.412·47-s − 0.707·50-s + 0.392·52-s − 1.92·53-s − 0.262·58-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: no
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 \( 1 + 5T^{2} \)
13 \( 1 - 2.82T + 13T^{2} \)
17 \( 1 - 2.82T + 17T^{2} \)
19 \( 1 + 5.65T + 19T^{2} \)
23 \( 1 + 8T + 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + 8.48T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 - 2.82T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 + 14T + 53T^{2} \)
59 \( 1 + 8.48T + 59T^{2} \)
61 \( 1 - 8.48T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 14.1T + 73T^{2} \)
79 \( 1 + 16T + 79T^{2} \)
83 \( 1 + 16.9T + 83T^{2} \)
89 \( 1 + 11.3T + 89T^{2} \)
97 \( 1 - 16.9T + 97T^{2} \)
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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

−7.26291126010817866608730322856, −6.45379480453288463630673707805, −5.91945577831179393514408992648, −5.44412349960301715849966381951, −4.31742103928807866958260303629, −3.94894170634214286509505704383, −3.23199485372711231632169723570, −2.12973997053598942616145080949, −1.53611668743012339964030891379, 0, 1.53611668743012339964030891379, 2.12973997053598942616145080949, 3.23199485372711231632169723570, 3.94894170634214286509505704383, 4.31742103928807866958260303629, 5.44412349960301715849966381951, 5.91945577831179393514408992648, 6.45379480453288463630673707805, 7.26291126010817866608730322856

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