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Properties

Label 2-60984-1.1-c1-0-25
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  − 2·5-s + 7-s − 2·13-s − 6·17-s − 8·19-s − 25-s + 6·29-s + 8·31-s − 2·35-s − 2·37-s + 2·41-s + 4·43-s + 8·47-s + 49-s − 6·53-s + 6·61-s + 4·65-s − 4·67-s + 8·71-s − 10·73-s − 16·79-s + 8·83-s + 12·85-s + 6·89-s − 2·91-s + 16·95-s − 6·97-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.377·7-s − 0.554·13-s − 1.45·17-s − 1.83·19-s − 1/5·25-s + 1.11·29-s + 1.43·31-s − 0.338·35-s − 0.328·37-s + 0.312·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s − 0.824·53-s + 0.768·61-s + 0.496·65-s − 0.488·67-s + 0.949·71-s − 1.17·73-s − 1.80·79-s + 0.878·83-s + 1.30·85-s + 0.635·89-s − 0.209·91-s + 1.64·95-s − 0.609·97-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 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−14.62297767985927, −14.07287784093089, −13.52440038408358, −13.03232830409515, −12.34390065271011, −12.14295007834627, −11.45276232431725, −11.06582659781582, −10.51090532278144, −10.13132888518979, −9.281144028618691, −8.808689438530008, −8.223782559149617, −8.029619443338096, −7.179467769718786, −6.756672933215835, −6.246230482620132, −5.557770527280153, −4.613149323413015, −4.412919304977472, −4.020871050842899, −2.997834451841972, −2.434277864283944, −1.845934023833014, −0.7484568076794646, 0, 0.7484568076794646, 1.845934023833014, 2.434277864283944, 2.997834451841972, 4.020871050842899, 4.412919304977472, 4.613149323413015, 5.557770527280153, 6.246230482620132, 6.756672933215835, 7.179467769718786, 8.029619443338096, 8.223782559149617, 8.808689438530008, 9.281144028618691, 10.13132888518979, 10.51090532278144, 11.06582659781582, 11.45276232431725, 12.14295007834627, 12.34390065271011, 13.03232830409515, 13.52440038408358, 14.07287784093089, 14.62297767985927

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