Properties

Label 2-60984-1.1-c1-0-53
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  − 5-s + 7-s + 4·13-s + 3·17-s + 8·19-s + 4·23-s − 4·25-s − 8·29-s − 10·31-s − 35-s − 2·37-s − 6·41-s + 43-s + 13·47-s + 49-s + 8·53-s − 3·59-s − 4·65-s + 7·67-s − 6·71-s + 6·73-s − 8·79-s − 83-s − 3·85-s − 3·89-s + 4·91-s − 8·95-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s + 1.10·13-s + 0.727·17-s + 1.83·19-s + 0.834·23-s − 4/5·25-s − 1.48·29-s − 1.79·31-s − 0.169·35-s − 0.328·37-s − 0.937·41-s + 0.152·43-s + 1.89·47-s + 1/7·49-s + 1.09·53-s − 0.390·59-s − 0.496·65-s + 0.855·67-s − 0.712·71-s + 0.702·73-s − 0.900·79-s − 0.109·83-s − 0.325·85-s − 0.317·89-s + 0.419·91-s − 0.820·95-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 \( 1 + T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 13 T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
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

−14.59425108909600, −13.95244063663954, −13.57333895007279, −13.16234494343583, −12.37391344882808, −12.06391171708175, −11.37515410818226, −11.13780365348363, −10.61685121910773, −9.899065421880846, −9.340522571378657, −8.954147773430250, −8.320154772893318, −7.709382642238576, −7.311200637840609, −6.934121264348789, −5.902400608147223, −5.446432308830015, −5.256829589522191, −4.141837370901965, −3.708487529766631, −3.318508963161736, −2.442986157232917, −1.512556133766309, −1.087816702867116, 0, 1.087816702867116, 1.512556133766309, 2.442986157232917, 3.318508963161736, 3.708487529766631, 4.141837370901965, 5.256829589522191, 5.446432308830015, 5.902400608147223, 6.934121264348789, 7.311200637840609, 7.709382642238576, 8.320154772893318, 8.954147773430250, 9.340522571378657, 9.899065421880846, 10.61685121910773, 11.13780365348363, 11.37515410818226, 12.06391171708175, 12.37391344882808, 13.16234494343583, 13.57333895007279, 13.95244063663954, 14.59425108909600

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