Properties

Label 2-9660-1.1-c1-0-37
Degree $2$
Conductor $9660$
Sign $1$
Analytic cond. $77.1354$
Root an. cond. $8.78268$
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  − 3-s + 5-s + 7-s + 9-s − 11-s + 4·13-s − 15-s + 6·17-s + 7·19-s − 21-s − 23-s + 25-s − 27-s − 6·29-s + 4·31-s + 33-s + 35-s − 2·37-s − 4·39-s + 9·41-s + 2·43-s + 45-s + 7·47-s + 49-s − 6·51-s + 5·53-s − 55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 1.10·13-s − 0.258·15-s + 1.45·17-s + 1.60·19-s − 0.218·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.718·31-s + 0.174·33-s + 0.169·35-s − 0.328·37-s − 0.640·39-s + 1.40·41-s + 0.304·43-s + 0.149·45-s + 1.02·47-s + 1/7·49-s − 0.840·51-s + 0.686·53-s − 0.134·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9660 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9660\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(77.1354\)
Root analytic conductor: \(8.78268\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{9660} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9660,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.472630130\)
\(L(\frac12)\) \(\approx\) \(2.472630130\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 - T \)
23 \( 1 + T \)
good11 \( 1 + T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 - 5 T + p T^{2} \)
59 \( 1 - 7 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
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

−7.64222071249777309420162100265, −7.07202991726494029092480482092, −6.07248405517259264181598087947, −5.62467582435781728943707867361, −5.21786473129594586744537208960, −4.18749598134104065704818762638, −3.48488895114235697623238538780, −2.61463233742959485888326091410, −1.45346245589439546542862235463, −0.873591127470956303153087243496, 0.873591127470956303153087243496, 1.45346245589439546542862235463, 2.61463233742959485888326091410, 3.48488895114235697623238538780, 4.18749598134104065704818762638, 5.21786473129594586744537208960, 5.62467582435781728943707867361, 6.07248405517259264181598087947, 7.07202991726494029092480482092, 7.64222071249777309420162100265

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