Properties

Label 2-9702-1.1-c1-0-154
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 8-s + 11-s + 2·13-s + 16-s − 4·17-s − 6·19-s + 22-s + 4·23-s − 5·25-s + 2·26-s + 10·29-s − 6·31-s + 32-s − 4·34-s − 6·37-s − 6·38-s − 12·41-s − 8·43-s + 44-s + 4·46-s + 2·47-s − 5·50-s + 2·52-s − 6·53-s + 10·58-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s + 0.301·11-s + 0.554·13-s + 1/4·16-s − 0.970·17-s − 1.37·19-s + 0.213·22-s + 0.834·23-s − 25-s + 0.392·26-s + 1.85·29-s − 1.07·31-s + 0.176·32-s − 0.685·34-s − 0.986·37-s − 0.973·38-s − 1.87·41-s − 1.21·43-s + 0.150·44-s + 0.589·46-s + 0.291·47-s − 0.707·50-s + 0.277·52-s − 0.824·53-s + 1.31·58-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: \(1\)
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 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−7.04911294487863644947847596171, −6.54457046719268389329282564708, −6.12928016763066447729285563880, −5.08316559941505302422469262039, −4.64702589496963019502504749433, −3.80292644671654860631338543879, −3.21020151266381282202645511235, −2.18842563100362433161385287823, −1.50131290588153234653626168607, 0, 1.50131290588153234653626168607, 2.18842563100362433161385287823, 3.21020151266381282202645511235, 3.80292644671654860631338543879, 4.64702589496963019502504749433, 5.08316559941505302422469262039, 6.12928016763066447729285563880, 6.54457046719268389329282564708, 7.04911294487863644947847596171

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