Properties

Label 2-9702-1.1-c1-0-156
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 + 1.41·5-s + 8-s + 1.41·10-s − 11-s − 5.65·13-s + 16-s − 1.41·17-s + 4.24·19-s + 1.41·20-s − 22-s − 4·23-s − 2.99·25-s − 5.65·26-s + 4.24·31-s + 32-s − 1.41·34-s − 6·37-s + 4.24·38-s + 1.41·40-s − 4.24·41-s − 4·43-s − 44-s − 4·46-s + 1.41·47-s − 2.99·50-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 0.632·5-s + 0.353·8-s + 0.447·10-s − 0.301·11-s − 1.56·13-s + 0.250·16-s − 0.342·17-s + 0.973·19-s + 0.316·20-s − 0.213·22-s − 0.834·23-s − 0.599·25-s − 1.10·26-s + 0.762·31-s + 0.176·32-s − 0.242·34-s − 0.986·37-s + 0.688·38-s + 0.223·40-s − 0.662·41-s − 0.609·43-s − 0.150·44-s − 0.589·46-s + 0.206·47-s − 0.424·50-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 \( 1 - 1.41T + 5T^{2} \)
13 \( 1 + 5.65T + 13T^{2} \)
17 \( 1 + 1.41T + 17T^{2} \)
19 \( 1 - 4.24T + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 4.24T + 31T^{2} \)
37 \( 1 + 6T + 37T^{2} \)
41 \( 1 + 4.24T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 - 1.41T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 - 2.82T + 59T^{2} \)
61 \( 1 - 5.65T + 61T^{2} \)
67 \( 1 + 10T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 4.24T + 73T^{2} \)
79 \( 1 + 6T + 79T^{2} \)
83 \( 1 + 15.5T + 83T^{2} \)
89 \( 1 + 14.1T + 89T^{2} \)
97 \( 1 - 5.65T + 97T^{2} \)
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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

−7.15864281472915593628486427273, −6.69621585909817045455723259335, −5.72327430547119134884953254611, −5.35829942180206146213679504852, −4.65385289146475982137414889001, −3.89716044657007891144087084576, −2.91921351851052192972769098485, −2.34357365138678474604547506364, −1.51628112424645981871697138837, 0, 1.51628112424645981871697138837, 2.34357365138678474604547506364, 2.91921351851052192972769098485, 3.89716044657007891144087084576, 4.65385289146475982137414889001, 5.35829942180206146213679504852, 5.72327430547119134884953254611, 6.69621585909817045455723259335, 7.15864281472915593628486427273

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