Properties

Label 2-60984-1.1-c1-0-0
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  + 5-s − 7-s − 3·13-s − 3·19-s − 8·23-s − 4·25-s − 7·29-s − 2·31-s − 35-s + 5·37-s − 4·41-s + 4·43-s + 3·47-s + 49-s + 6·53-s + 3·59-s − 10·61-s − 3·65-s + 5·67-s − 12·71-s − 5·73-s − 10·79-s + 12·83-s − 6·89-s + 3·91-s − 3·95-s + 12·97-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s − 0.832·13-s − 0.688·19-s − 1.66·23-s − 4/5·25-s − 1.29·29-s − 0.359·31-s − 0.169·35-s + 0.821·37-s − 0.624·41-s + 0.609·43-s + 0.437·47-s + 1/7·49-s + 0.824·53-s + 0.390·59-s − 1.28·61-s − 0.372·65-s + 0.610·67-s − 1.42·71-s − 0.585·73-s − 1.12·79-s + 1.31·83-s − 0.635·89-s + 0.314·91-s − 0.307·95-s + 1.21·97-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

\(L(1)\) \(\approx\) \(0.7290864444\)
\(L(\frac12)\) \(\approx\) \(0.7290864444\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 5 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−14.28791958501270, −13.70083385624037, −13.37070801832398, −12.77236937044508, −12.30305027609614, −11.82708760723559, −11.30632330952676, −10.60557814865513, −10.16132486103764, −9.708114777455404, −9.270840426915563, −8.691838473185699, −7.995514194963042, −7.529326845947281, −7.037922742254569, −6.266587694043411, −5.848127080667430, −5.452118048185485, −4.552382003037352, −4.077134292429895, −3.503621372349085, −2.580039038888765, −2.149810495949778, −1.495909902314398, −0.2738947975451116, 0.2738947975451116, 1.495909902314398, 2.149810495949778, 2.580039038888765, 3.503621372349085, 4.077134292429895, 4.552382003037352, 5.452118048185485, 5.848127080667430, 6.266587694043411, 7.037922742254569, 7.529326845947281, 7.995514194963042, 8.691838473185699, 9.270840426915563, 9.708114777455404, 10.16132486103764, 10.60557814865513, 11.30632330952676, 11.82708760723559, 12.30305027609614, 12.77236937044508, 13.37070801832398, 13.70083385624037, 14.28791958501270

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