Properties

Label 2-60984-1.1-c1-0-1
Degree $2$
Conductor $60984$
Sign $1$
Analytic cond. $486.959$
Root an. cond. $22.0671$
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  − 7-s − 2·13-s − 8·17-s + 6·19-s − 8·23-s − 5·25-s − 6·29-s + 10·31-s + 2·37-s − 8·41-s + 4·43-s − 2·47-s + 49-s − 2·53-s + 8·59-s − 2·61-s + 4·67-s + 4·71-s + 4·73-s + 8·79-s − 18·83-s − 2·89-s + 2·91-s − 14·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 0.377·7-s − 0.554·13-s − 1.94·17-s + 1.37·19-s − 1.66·23-s − 25-s − 1.11·29-s + 1.79·31-s + 0.328·37-s − 1.24·41-s + 0.609·43-s − 0.291·47-s + 1/7·49-s − 0.274·53-s + 1.04·59-s − 0.256·61-s + 0.488·67-s + 0.474·71-s + 0.468·73-s + 0.900·79-s − 1.97·83-s − 0.211·89-s + 0.209·91-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60984\)    =    \(2^{3} \cdot 3^{2} \cdot 7 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(486.959\)
Root analytic conductor: \(22.0671\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 60984,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7228321163\)
\(L(\frac12)\) \(\approx\) \(0.7228321163\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + T \)
11 \( 1 \)
good5 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 18 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−14.01311546635561, −13.82843002055587, −13.41929070756521, −12.80933310237664, −12.19836339181941, −11.75662862142621, −11.35154826345109, −10.80033159569254, −9.953642278430029, −9.774419359571346, −9.341891150937885, −8.502086889072131, −8.147720152092535, −7.523300821676577, −6.913540571960356, −6.474759174039291, −5.840331431273919, −5.315954237034663, −4.572991645529142, −4.076434939532783, −3.505406111280672, −2.608263672964768, −2.215099766165204, −1.394940176292454, −0.2819100172794180, 0.2819100172794180, 1.394940176292454, 2.215099766165204, 2.608263672964768, 3.505406111280672, 4.076434939532783, 4.572991645529142, 5.315954237034663, 5.840331431273919, 6.474759174039291, 6.913540571960356, 7.523300821676577, 8.147720152092535, 8.502086889072131, 9.341891150937885, 9.774419359571346, 9.953642278430029, 10.80033159569254, 11.35154826345109, 11.75662862142621, 12.19836339181941, 12.80933310237664, 13.41929070756521, 13.82843002055587, 14.01311546635561

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