Properties

Label 2-9702-1.1-c1-0-143
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 + 1.61·5-s − 8-s − 1.61·10-s + 11-s + 16-s + 7.34·17-s + 1.17·19-s + 1.61·20-s − 22-s − 3.55·23-s − 2.39·25-s − 10.3·29-s + 6.34·31-s − 32-s − 7.34·34-s + 2.82·37-s − 1.17·38-s − 1.61·40-s − 4.94·41-s − 11.5·43-s + 44-s + 3.55·46-s − 6.78·47-s + 2.39·50-s − 8·53-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 0.721·5-s − 0.353·8-s − 0.510·10-s + 0.301·11-s + 0.250·16-s + 1.78·17-s + 0.268·19-s + 0.360·20-s − 0.213·22-s − 0.741·23-s − 0.479·25-s − 1.93·29-s + 1.13·31-s − 0.176·32-s − 1.25·34-s + 0.465·37-s − 0.189·38-s − 0.255·40-s − 0.772·41-s − 1.75·43-s + 0.150·44-s + 0.524·46-s − 0.989·47-s + 0.339·50-s − 1.09·53-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: \(1\)
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 \( 1 - 1.61T + 5T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 7.34T + 17T^{2} \)
19 \( 1 - 1.17T + 19T^{2} \)
23 \( 1 + 3.55T + 23T^{2} \)
29 \( 1 + 10.3T + 29T^{2} \)
31 \( 1 - 6.34T + 31T^{2} \)
37 \( 1 - 2.82T + 37T^{2} \)
41 \( 1 + 4.94T + 41T^{2} \)
43 \( 1 + 11.5T + 43T^{2} \)
47 \( 1 + 6.78T + 47T^{2} \)
53 \( 1 + 8T + 53T^{2} \)
59 \( 1 + 4.39T + 59T^{2} \)
61 \( 1 + 11.9T + 61T^{2} \)
67 \( 1 - 2.94T + 67T^{2} \)
71 \( 1 + 15.5T + 71T^{2} \)
73 \( 1 + 5.11T + 73T^{2} \)
79 \( 1 - 5.61T + 79T^{2} \)
83 \( 1 + 5.05T + 83T^{2} \)
89 \( 1 - 0.773T + 89T^{2} \)
97 \( 1 - 7T + 97T^{2} \)
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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

−7.58706112006155892498588354682, −6.64802767496475947553437890589, −5.99440362039303322292340427587, −5.54224113410139393507225843392, −4.65303737481324065232243151987, −3.55821888774025826215802508992, −3.01080496136979650057679337074, −1.82692573456607302970413694887, −1.39126790606822179345833944557, 0, 1.39126790606822179345833944557, 1.82692573456607302970413694887, 3.01080496136979650057679337074, 3.55821888774025826215802508992, 4.65303737481324065232243151987, 5.54224113410139393507225843392, 5.99440362039303322292340427587, 6.64802767496475947553437890589, 7.58706112006155892498588354682

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