Properties

Label 2-60984-1.1-c1-0-5
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 + 13-s + 8·17-s + 5·19-s − 4·25-s − 5·29-s − 6·31-s + 35-s − 11·37-s + 8·43-s − 7·47-s + 49-s − 2·53-s − 7·59-s − 2·61-s − 65-s − 7·67-s + 7·73-s + 2·79-s + 12·83-s − 8·85-s − 2·89-s − 91-s − 5·95-s − 8·97-s + 101-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s + 0.277·13-s + 1.94·17-s + 1.14·19-s − 4/5·25-s − 0.928·29-s − 1.07·31-s + 0.169·35-s − 1.80·37-s + 1.21·43-s − 1.02·47-s + 1/7·49-s − 0.274·53-s − 0.911·59-s − 0.256·61-s − 0.124·65-s − 0.855·67-s + 0.819·73-s + 0.225·79-s + 1.31·83-s − 0.867·85-s − 0.211·89-s − 0.104·91-s − 0.512·95-s − 0.812·97-s + 0.0995·101-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: \(0\)
Selberg data: \((2,\ 60984,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.645774393\)
\(L(\frac12)\) \(\approx\) \(1.645774393\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 7 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−14.24123117099723, −13.80072870301369, −13.40623638634793, −12.55849256306412, −12.30952455176430, −11.89773522379373, −11.22697782144499, −10.80218959166026, −10.16939311241642, −9.588327455305590, −9.343497363001768, −8.596810121880410, −7.927443267035831, −7.530137829228351, −7.188942184651927, −6.374652176554629, −5.616409305374403, −5.498147520078140, −4.696176619254396, −3.806669457895306, −3.442532176669161, −3.055197030603400, −1.947052469252081, −1.349852131986831, −0.4528043344424230, 0.4528043344424230, 1.349852131986831, 1.947052469252081, 3.055197030603400, 3.442532176669161, 3.806669457895306, 4.696176619254396, 5.498147520078140, 5.616409305374403, 6.374652176554629, 7.188942184651927, 7.530137829228351, 7.927443267035831, 8.596810121880410, 9.343497363001768, 9.588327455305590, 10.16939311241642, 10.80218959166026, 11.22697782144499, 11.89773522379373, 12.30952455176430, 12.55849256306412, 13.40623638634793, 13.80072870301369, 14.24123117099723

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