Properties

Label 2-9702-1.1-c1-0-14
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 0.585·5-s − 8-s − 0.585·10-s − 11-s − 3.82·13-s + 16-s − 3.65·17-s − 0.585·19-s + 0.585·20-s + 22-s + 6.24·23-s − 4.65·25-s + 3.82·26-s − 2.65·29-s − 4·31-s − 32-s + 3.65·34-s − 9.41·37-s + 0.585·38-s − 0.585·40-s + 5.41·41-s − 5.65·43-s − 44-s − 6.24·46-s + 10.4·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 0.261·5-s − 0.353·8-s − 0.185·10-s − 0.301·11-s − 1.06·13-s + 0.250·16-s − 0.886·17-s − 0.134·19-s + 0.130·20-s + 0.213·22-s + 1.30·23-s − 0.931·25-s + 0.750·26-s − 0.493·29-s − 0.718·31-s − 0.176·32-s + 0.627·34-s − 1.54·37-s + 0.0950·38-s − 0.0926·40-s + 0.845·41-s − 0.862·43-s − 0.150·44-s − 0.920·46-s + 1.52·47-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: no
Arithmetic: yes
Character: $\chi_{9702} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9702,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9414401584\)
\(L(\frac12)\) \(\approx\) \(0.9414401584\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.585T + 5T^{2} \)
13 \( 1 + 3.82T + 13T^{2} \)
17 \( 1 + 3.65T + 17T^{2} \)
19 \( 1 + 0.585T + 19T^{2} \)
23 \( 1 - 6.24T + 23T^{2} \)
29 \( 1 + 2.65T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + 9.41T + 37T^{2} \)
41 \( 1 - 5.41T + 41T^{2} \)
43 \( 1 + 5.65T + 43T^{2} \)
47 \( 1 - 10.4T + 47T^{2} \)
53 \( 1 + 7.89T + 53T^{2} \)
59 \( 1 - 5.58T + 59T^{2} \)
61 \( 1 - 11.8T + 61T^{2} \)
67 \( 1 - 2.75T + 67T^{2} \)
71 \( 1 - 11.0T + 71T^{2} \)
73 \( 1 + 9.41T + 73T^{2} \)
79 \( 1 + 13.2T + 79T^{2} \)
83 \( 1 - 12.1T + 83T^{2} \)
89 \( 1 + 12.4T + 89T^{2} \)
97 \( 1 + 3.82T + 97T^{2} \)
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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

−7.51183389471990926566209118894, −7.19576411792399136382220659439, −6.51882453803484115696885308398, −5.60378833607157166717260971034, −5.09245245511981730746161516684, −4.18720382324560450632702719503, −3.25167360235717898908621419292, −2.36802872553764179731784778409, −1.80070410638963361008062689809, −0.49362482980919079971154421053, 0.49362482980919079971154421053, 1.80070410638963361008062689809, 2.36802872553764179731784778409, 3.25167360235717898908621419292, 4.18720382324560450632702719503, 5.09245245511981730746161516684, 5.60378833607157166717260971034, 6.51882453803484115696885308398, 7.19576411792399136382220659439, 7.51183389471990926566209118894

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