Properties

Label 2-9702-1.1-c1-0-11
Degree $2$
Conductor $9702$
Sign $1$
Analytic cond. $77.4708$
Root an. cond. $8.80175$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

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 − 2·13-s + 16-s + 17-s + 3·19-s − 2·20-s + 22-s + 23-s − 25-s + 2·26-s + 29-s + 2·31-s − 32-s − 34-s − 5·37-s − 3·38-s + 2·40-s − 10·41-s + 43-s − 44-s − 46-s + 7·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.353·8-s + 0.632·10-s − 0.301·11-s − 0.554·13-s + 1/4·16-s + 0.242·17-s + 0.688·19-s − 0.447·20-s + 0.213·22-s + 0.208·23-s − 1/5·25-s + 0.392·26-s + 0.185·29-s + 0.359·31-s − 0.176·32-s − 0.171·34-s − 0.821·37-s − 0.486·38-s + 0.316·40-s − 1.56·41-s + 0.152·43-s − 0.150·44-s − 0.147·46-s + 1.02·47-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: yes
Arithmetic: yes
Character: $\chi_{9702} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9702,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7757359258\)
\(L(\frac12)\) \(\approx\) \(0.7757359258\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 - 3 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−7.85747770802078815545372392454, −7.07460183022210449527023211966, −6.65765666986493420660734961996, −5.56720138691284046709681890077, −5.02902822390966345594923202582, −4.07525073211865380356777462168, −3.34421680996131307135244884602, −2.59569625731930446489042912023, −1.56672594819181568856264595472, −0.47141604315454646824999418909, 0.47141604315454646824999418909, 1.56672594819181568856264595472, 2.59569625731930446489042912023, 3.34421680996131307135244884602, 4.07525073211865380356777462168, 5.02902822390966345594923202582, 5.56720138691284046709681890077, 6.65765666986493420660734961996, 7.07460183022210449527023211966, 7.85747770802078815545372392454

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