Properties

Label 2-9702-1.1-c1-0-3
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 − 2.47·5-s − 8-s + 2.47·10-s − 11-s − 3.03·13-s + 16-s − 6.64·17-s − 0.557·19-s − 2.47·20-s + 22-s + 2.66·23-s + 1.11·25-s + 3.03·26-s − 3.77·29-s − 2.00·31-s − 32-s + 6.64·34-s − 0.158·37-s + 0.557·38-s + 2.47·40-s − 5.93·41-s + 4.32·43-s − 44-s − 2.66·46-s − 7.38·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 1.10·5-s − 0.353·8-s + 0.782·10-s − 0.301·11-s − 0.840·13-s + 0.250·16-s − 1.61·17-s − 0.127·19-s − 0.553·20-s + 0.213·22-s + 0.556·23-s + 0.223·25-s + 0.594·26-s − 0.701·29-s − 0.360·31-s − 0.176·32-s + 1.13·34-s − 0.0260·37-s + 0.0904·38-s + 0.391·40-s − 0.926·41-s + 0.659·43-s − 0.150·44-s − 0.393·46-s − 1.07·47-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

\(L(1)\) \(\approx\) \(0.2655469127\)
\(L(\frac12)\) \(\approx\) \(0.2655469127\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.47T + 5T^{2} \)
13 \( 1 + 3.03T + 13T^{2} \)
17 \( 1 + 6.64T + 17T^{2} \)
19 \( 1 + 0.557T + 19T^{2} \)
23 \( 1 - 2.66T + 23T^{2} \)
29 \( 1 + 3.77T + 29T^{2} \)
31 \( 1 + 2.00T + 31T^{2} \)
37 \( 1 + 0.158T + 37T^{2} \)
41 \( 1 + 5.93T + 41T^{2} \)
43 \( 1 - 4.32T + 43T^{2} \)
47 \( 1 + 7.38T + 47T^{2} \)
53 \( 1 + 2.83T + 53T^{2} \)
59 \( 1 + 1.48T + 59T^{2} \)
61 \( 1 + 3.19T + 61T^{2} \)
67 \( 1 + 11.2T + 67T^{2} \)
71 \( 1 - 11.0T + 71T^{2} \)
73 \( 1 + 3.18T + 73T^{2} \)
79 \( 1 + 6.95T + 79T^{2} \)
83 \( 1 - 4.59T + 83T^{2} \)
89 \( 1 + 6.62T + 89T^{2} \)
97 \( 1 - 1.17T + 97T^{2} \)
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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

−7.79039412037028850243547758288, −7.06062059584348111931958548131, −6.68454492125963656736192929203, −5.67693596154000315968146650963, −4.80905643267898948794957069397, −4.21995658011911024667627159144, −3.33641082057976988580846083004, −2.53190155302618950413180578835, −1.67065542604690892796760911574, −0.26329443169943836493114912312, 0.26329443169943836493114912312, 1.67065542604690892796760911574, 2.53190155302618950413180578835, 3.33641082057976988580846083004, 4.21995658011911024667627159144, 4.80905643267898948794957069397, 5.67693596154000315968146650963, 6.68454492125963656736192929203, 7.06062059584348111931958548131, 7.79039412037028850243547758288

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