Properties

Label 2-60984-1.1-c1-0-39
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  − 7-s + 2·17-s + 4·19-s − 4·23-s − 5·25-s − 6·29-s + 4·31-s + 6·37-s + 6·41-s + 4·43-s − 12·47-s + 49-s + 4·53-s − 8·61-s + 12·67-s + 4·71-s + 4·73-s − 4·83-s + 12·89-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 2·119-s + ⋯
L(s)  = 1  − 0.377·7-s + 0.485·17-s + 0.917·19-s − 0.834·23-s − 25-s − 1.11·29-s + 0.718·31-s + 0.986·37-s + 0.937·41-s + 0.609·43-s − 1.75·47-s + 1/7·49-s + 0.549·53-s − 1.02·61-s + 1.46·67-s + 0.474·71-s + 0.468·73-s − 0.439·83-s + 1.27·89-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 0.183·119-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 \( 1 + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−14.49284823033570, −14.03305741890789, −13.56733217936818, −13.00401712343349, −12.61519745347785, −11.87378124168215, −11.65662981173203, −11.03545431712424, −10.43670996297599, −9.770207101438334, −9.564737204681689, −9.058608436136889, −8.144161110681061, −7.815434807616716, −7.430509914530488, −6.508672479927287, −6.241136803571127, −5.466538207895440, −5.158964966825104, −4.117844702630346, −3.885090305505909, −3.070802653929850, −2.489317902165030, −1.705566109190350, −0.9203624637792201, 0, 0.9203624637792201, 1.705566109190350, 2.489317902165030, 3.070802653929850, 3.885090305505909, 4.117844702630346, 5.158964966825104, 5.466538207895440, 6.241136803571127, 6.508672479927287, 7.430509914530488, 7.815434807616716, 8.144161110681061, 9.058608436136889, 9.564737204681689, 9.770207101438334, 10.43670996297599, 11.03545431712424, 11.65662981173203, 11.87378124168215, 12.61519745347785, 13.00401712343349, 13.56733217936818, 14.03305741890789, 14.49284823033570

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