Properties

Label 2-60984-1.1-c1-0-50
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·5-s − 7-s + 13-s − 2·17-s − 7·19-s + 4·23-s + 4·25-s + 7·29-s + 2·31-s − 3·35-s + 37-s − 12·41-s + 12·43-s + 9·47-s + 49-s − 12·53-s − 3·59-s − 10·61-s + 3·65-s − 7·67-s + 12·71-s + 15·73-s − 12·79-s − 18·83-s − 6·85-s + 10·89-s − 91-s + ⋯
L(s)  = 1  + 1.34·5-s − 0.377·7-s + 0.277·13-s − 0.485·17-s − 1.60·19-s + 0.834·23-s + 4/5·25-s + 1.29·29-s + 0.359·31-s − 0.507·35-s + 0.164·37-s − 1.87·41-s + 1.82·43-s + 1.31·47-s + 1/7·49-s − 1.64·53-s − 0.390·59-s − 1.28·61-s + 0.372·65-s − 0.855·67-s + 1.42·71-s + 1.75·73-s − 1.35·79-s − 1.97·83-s − 0.650·85-s + 1.05·89-s − 0.104·91-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: $\chi_{60984} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 60984,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 \)
7 \( 1 + T \)
11 \( 1 \)
good5 \( 1 - 3 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 15 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 18 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−14.34418818520982, −13.93307843370969, −13.65783203252875, −13.02844577165322, −12.57553484408605, −12.29527011252554, −11.35212671216084, −10.88662508556208, −10.41268730633080, −10.03841670245176, −9.367687169122008, −8.949323133537123, −8.537657040333959, −7.864022710784472, −7.065501481578540, −6.466899362543977, −6.272993284594519, −5.676940294070784, −4.935769077018918, −4.477999976267206, −3.760050392918791, −2.848519106736580, −2.490972446575786, −1.746565557555185, −1.073747636086400, 0, 1.073747636086400, 1.746565557555185, 2.490972446575786, 2.848519106736580, 3.760050392918791, 4.477999976267206, 4.935769077018918, 5.676940294070784, 6.272993284594519, 6.466899362543977, 7.065501481578540, 7.864022710784472, 8.537657040333959, 8.949323133537123, 9.367687169122008, 10.03841670245176, 10.41268730633080, 10.88662508556208, 11.35212671216084, 12.29527011252554, 12.57553484408605, 13.02844577165322, 13.65783203252875, 13.93307843370969, 14.34418818520982

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