Properties

Label 2-9702-1.1-c1-0-145
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 − 6·13-s + 16-s + 4·17-s − 6·19-s + 22-s + 4·23-s − 5·25-s − 6·26-s − 6·29-s + 2·31-s + 32-s + 4·34-s + 10·37-s − 6·38-s − 4·41-s + 8·43-s + 44-s + 4·46-s − 6·47-s − 5·50-s − 6·52-s + 10·53-s − 6·58-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s + 0.301·11-s − 1.66·13-s + 1/4·16-s + 0.970·17-s − 1.37·19-s + 0.213·22-s + 0.834·23-s − 25-s − 1.17·26-s − 1.11·29-s + 0.359·31-s + 0.176·32-s + 0.685·34-s + 1.64·37-s − 0.973·38-s − 0.624·41-s + 1.21·43-s + 0.150·44-s + 0.589·46-s − 0.875·47-s − 0.707·50-s − 0.832·52-s + 1.37·53-s − 0.787·58-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 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−7.41221070946741716285556701987, −6.59594076556384926061060344162, −5.84307339086204440003661386330, −5.32590227718596198771944680311, −4.42672463347886724354744044498, −4.04569326868045934263308666900, −2.95026189316440015396473989179, −2.39650278835518714351632828598, −1.41836430213139596112613988844, 0, 1.41836430213139596112613988844, 2.39650278835518714351632828598, 2.95026189316440015396473989179, 4.04569326868045934263308666900, 4.42672463347886724354744044498, 5.32590227718596198771944680311, 5.84307339086204440003661386330, 6.59594076556384926061060344162, 7.41221070946741716285556701987

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