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Properties

Label 2-60984-1.1-c1-0-27
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 + 3·13-s − 6·17-s − 19-s − 4·23-s + 4·25-s − 5·29-s − 4·31-s − 3·35-s + 5·37-s − 2·41-s − 6·43-s − 47-s + 49-s − 2·53-s + 7·59-s + 14·61-s − 9·65-s + 67-s + 6·71-s + 17·73-s − 2·79-s − 14·83-s + 18·85-s − 6·89-s + 3·91-s + ⋯
L(s)  = 1  − 1.34·5-s + 0.377·7-s + 0.832·13-s − 1.45·17-s − 0.229·19-s − 0.834·23-s + 4/5·25-s − 0.928·29-s − 0.718·31-s − 0.507·35-s + 0.821·37-s − 0.312·41-s − 0.914·43-s − 0.145·47-s + 1/7·49-s − 0.274·53-s + 0.911·59-s + 1.79·61-s − 1.11·65-s + 0.122·67-s + 0.712·71-s + 1.98·73-s − 0.225·79-s − 1.53·83-s + 1.95·85-s − 0.635·89-s + 0.314·91-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: Trivial
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 \( 1 - 3 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 7 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 17 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−14.76647132177069, −14.01396669678573, −13.53950341896442, −12.90673437046647, −12.64916903695682, −11.82620759273619, −11.44796648647582, −11.10441886762126, −10.77278960947689, −9.882744870613372, −9.443104516003864, −8.566419829356948, −8.406883687419709, −7.964780956480359, −7.150284372718704, −6.907976937259026, −6.141946314743141, −5.554647689848893, −4.805263731904084, −4.246518448304685, −3.802117765268705, −3.350768354760315, −2.306337720330293, −1.818161567894596, −0.7516659395404280, 0, 0.7516659395404280, 1.818161567894596, 2.306337720330293, 3.350768354760315, 3.802117765268705, 4.246518448304685, 4.805263731904084, 5.554647689848893, 6.141946314743141, 6.907976937259026, 7.150284372718704, 7.964780956480359, 8.406883687419709, 8.566419829356948, 9.443104516003864, 9.882744870613372, 10.77278960947689, 11.10441886762126, 11.44796648647582, 11.82620759273619, 12.64916903695682, 12.90673437046647, 13.53950341896442, 14.01396669678573, 14.76647132177069

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