Properties

Label 2-9702-1.1-c1-0-107
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 − 2·5-s − 8-s + 2·10-s + 11-s + 5.41·13-s + 16-s − 2·17-s − 2.24·19-s − 2·20-s − 22-s + 4.82·23-s − 25-s − 5.41·26-s − 9.65·29-s + 6.24·31-s − 32-s + 2·34-s + 0.828·37-s + 2.24·38-s + 2·40-s − 11.6·41-s − 4.82·43-s + 44-s − 4.82·46-s + 7.89·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 0.894·5-s − 0.353·8-s + 0.632·10-s + 0.301·11-s + 1.50·13-s + 0.250·16-s − 0.485·17-s − 0.514·19-s − 0.447·20-s − 0.213·22-s + 1.00·23-s − 0.200·25-s − 1.06·26-s − 1.79·29-s + 1.12·31-s − 0.176·32-s + 0.342·34-s + 0.136·37-s + 0.363·38-s + 0.316·40-s − 1.82·41-s − 0.736·43-s + 0.150·44-s − 0.711·46-s + 1.15·47-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: 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 \( 1 + 2T + 5T^{2} \)
13 \( 1 - 5.41T + 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + 2.24T + 19T^{2} \)
23 \( 1 - 4.82T + 23T^{2} \)
29 \( 1 + 9.65T + 29T^{2} \)
31 \( 1 - 6.24T + 31T^{2} \)
37 \( 1 - 0.828T + 37T^{2} \)
41 \( 1 + 11.6T + 41T^{2} \)
43 \( 1 + 4.82T + 43T^{2} \)
47 \( 1 - 7.89T + 47T^{2} \)
53 \( 1 - 10.4T + 53T^{2} \)
59 \( 1 + 6.82T + 59T^{2} \)
61 \( 1 - 2.58T + 61T^{2} \)
67 \( 1 + 1.17T + 67T^{2} \)
71 \( 1 + 5.65T + 71T^{2} \)
73 \( 1 - 11.6T + 73T^{2} \)
79 \( 1 + 16.4T + 79T^{2} \)
83 \( 1 + 5.07T + 83T^{2} \)
89 \( 1 + 3.75T + 89T^{2} \)
97 \( 1 + 9.89T + 97T^{2} \)
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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

−7.34631135086647411826343601970, −6.84398101536375019088910068645, −6.14100495040915870233285696474, −5.41473216242024660065866511989, −4.35313128952855843786545992903, −3.77095578713166586613405112466, −3.09015424304817645981087603647, −1.96894898700370322778528809264, −1.09552030927412792905760582792, 0, 1.09552030927412792905760582792, 1.96894898700370322778528809264, 3.09015424304817645981087603647, 3.77095578713166586613405112466, 4.35313128952855843786545992903, 5.41473216242024660065866511989, 6.14100495040915870233285696474, 6.84398101536375019088910068645, 7.34631135086647411826343601970

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