Properties

Label 2-17600-1.1-c1-0-7
Degree $2$
Conductor $17600$
Sign $1$
Analytic cond. $140.536$
Root an. cond. $11.8548$
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  − 3·9-s − 11-s + 2·13-s − 6·17-s − 4·19-s + 4·23-s − 6·29-s + 8·31-s − 2·37-s + 2·41-s − 4·43-s − 12·47-s − 7·49-s − 2·53-s + 4·59-s + 10·61-s + 16·67-s − 8·71-s − 14·73-s − 8·79-s + 9·81-s + 4·83-s + 10·89-s − 10·97-s + 3·99-s + 101-s + 103-s + ⋯
L(s)  = 1  − 9-s − 0.301·11-s + 0.554·13-s − 1.45·17-s − 0.917·19-s + 0.834·23-s − 1.11·29-s + 1.43·31-s − 0.328·37-s + 0.312·41-s − 0.609·43-s − 1.75·47-s − 49-s − 0.274·53-s + 0.520·59-s + 1.28·61-s + 1.95·67-s − 0.949·71-s − 1.63·73-s − 0.900·79-s + 81-s + 0.439·83-s + 1.05·89-s − 1.01·97-s + 0.301·99-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17600\)    =    \(2^{6} \cdot 5^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(140.536\)
Root analytic conductor: \(11.8548\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 17600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.088628914\)
\(L(\frac12)\) \(\approx\) \(1.088628914\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
11 \( 1 + T \)
good3 \( 1 + p T^{2} \)
7 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
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

−15.91022624964817, −15.17983219015281, −14.80243011511883, −14.26510568353667, −13.47334713241328, −13.14271820390713, −12.72114885513424, −11.75096141847157, −11.24589707648236, −11.07570315400737, −10.24092905422071, −9.649991993412616, −8.819375928911702, −8.546677207409666, −8.029423838814662, −7.103515319526206, −6.485229438804265, −6.088592828738763, −5.201913089263542, −4.702833518494879, −3.890483565950169, −3.127801226680374, −2.444029238442218, −1.681506284470787, −0.4244767051377569, 0.4244767051377569, 1.681506284470787, 2.444029238442218, 3.127801226680374, 3.890483565950169, 4.702833518494879, 5.201913089263542, 6.088592828738763, 6.485229438804265, 7.103515319526206, 8.029423838814662, 8.546677207409666, 8.819375928911702, 9.649991993412616, 10.24092905422071, 11.07570315400737, 11.24589707648236, 11.75096141847157, 12.72114885513424, 13.14271820390713, 13.47334713241328, 14.26510568353667, 14.80243011511883, 15.17983219015281, 15.91022624964817

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