Properties

Label 2-60984-1.1-c1-0-36
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 − 3·13-s + 2·17-s + 5·19-s + 8·23-s − 4·25-s + 7·29-s − 8·31-s + 35-s − 3·37-s − 10·41-s + 10·43-s − 7·47-s + 49-s + 2·53-s + 9·59-s + 2·61-s + 3·65-s − 3·67-s − 6·71-s − 73-s − 10·79-s − 6·83-s − 2·85-s − 2·89-s + 3·91-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s − 0.832·13-s + 0.485·17-s + 1.14·19-s + 1.66·23-s − 4/5·25-s + 1.29·29-s − 1.43·31-s + 0.169·35-s − 0.493·37-s − 1.56·41-s + 1.52·43-s − 1.02·47-s + 1/7·49-s + 0.274·53-s + 1.17·59-s + 0.256·61-s + 0.372·65-s − 0.366·67-s − 0.712·71-s − 0.117·73-s − 1.12·79-s − 0.658·83-s − 0.216·85-s − 0.211·89-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: $\chi_{60984} (1, \cdot )$
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 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
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

−14.46392486121947, −14.16634982403406, −13.50388927321314, −12.97379025161474, −12.53896573932557, −11.99248706802022, −11.55005219777013, −11.13717630786397, −10.30040858579209, −10.03398643274454, −9.444827343443013, −8.887299502719509, −8.411294580811502, −7.648481275508674, −7.223745976602218, −6.933590272418969, −6.103277470696864, −5.398537796534604, −5.076953520694589, −4.375996021173067, −3.614104653795494, −3.147197486025564, −2.588588164913750, −1.645200312231988, −0.8791320798993977, 0, 0.8791320798993977, 1.645200312231988, 2.588588164913750, 3.147197486025564, 3.614104653795494, 4.375996021173067, 5.076953520694589, 5.398537796534604, 6.103277470696864, 6.933590272418969, 7.223745976602218, 7.648481275508674, 8.411294580811502, 8.887299502719509, 9.444827343443013, 10.03398643274454, 10.30040858579209, 11.13717630786397, 11.55005219777013, 11.99248706802022, 12.53896573932557, 12.97379025161474, 13.50388927321314, 14.16634982403406, 14.46392486121947

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