Properties

Label 2-9702-1.1-c1-0-167
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 + 2.64·5-s + 8-s + 2.64·10-s − 11-s − 4·13-s + 16-s + 3·17-s − 5.29·19-s + 2.64·20-s − 22-s − 2.64·23-s + 2.00·25-s − 4·26-s − 2·29-s − 4·31-s + 32-s + 3·34-s − 9.29·37-s − 5.29·38-s + 2.64·40-s − 9·41-s − 1.29·43-s − 44-s − 2.64·46-s − 11.9·47-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 1.18·5-s + 0.353·8-s + 0.836·10-s − 0.301·11-s − 1.10·13-s + 0.250·16-s + 0.727·17-s − 1.21·19-s + 0.591·20-s − 0.213·22-s − 0.551·23-s + 0.400·25-s − 0.784·26-s − 0.371·29-s − 0.718·31-s + 0.176·32-s + 0.514·34-s − 1.52·37-s − 0.858·38-s + 0.418·40-s − 1.40·41-s − 0.196·43-s − 0.150·44-s − 0.390·46-s − 1.74·47-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 \( 1 - 2.64T + 5T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 + 5.29T + 19T^{2} \)
23 \( 1 + 2.64T + 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + 9.29T + 37T^{2} \)
41 \( 1 + 9T + 41T^{2} \)
43 \( 1 + 1.29T + 43T^{2} \)
47 \( 1 + 11.9T + 47T^{2} \)
53 \( 1 + 4T + 53T^{2} \)
59 \( 1 - 14.5T + 59T^{2} \)
61 \( 1 + 3.93T + 61T^{2} \)
67 \( 1 + 13.5T + 67T^{2} \)
71 \( 1 - 13.2T + 71T^{2} \)
73 \( 1 + 4.70T + 73T^{2} \)
79 \( 1 - 5.35T + 79T^{2} \)
83 \( 1 - 5.58T + 83T^{2} \)
89 \( 1 + 13.2T + 89T^{2} \)
97 \( 1 - 9.58T + 97T^{2} \)
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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

−7.10526583277237411484371916517, −6.57701248164210012835587204339, −5.85317841156047568771201320891, −5.25832770121441114498776103726, −4.79588018584106298209245175258, −3.78526277519252135488512400289, −3.03379175806038509593832848952, −2.07583230112106834888351709169, −1.73059376129159953175002230519, 0, 1.73059376129159953175002230519, 2.07583230112106834888351709169, 3.03379175806038509593832848952, 3.78526277519252135488512400289, 4.79588018584106298209245175258, 5.25832770121441114498776103726, 5.85317841156047568771201320891, 6.57701248164210012835587204339, 7.10526583277237411484371916517

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