Properties

Label 2-60984-1.1-c1-0-44
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 + 6·13-s − 2·17-s + 8·19-s + 4·23-s − 25-s + 2·29-s − 8·31-s + 2·35-s + 6·37-s − 2·41-s − 8·43-s + 4·47-s + 49-s − 2·53-s + 12·59-s − 10·61-s − 12·65-s − 12·67-s + 12·71-s − 10·73-s + 8·79-s + 12·83-s + 4·85-s − 10·89-s − 6·91-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.377·7-s + 1.66·13-s − 0.485·17-s + 1.83·19-s + 0.834·23-s − 1/5·25-s + 0.371·29-s − 1.43·31-s + 0.338·35-s + 0.986·37-s − 0.312·41-s − 1.21·43-s + 0.583·47-s + 1/7·49-s − 0.274·53-s + 1.56·59-s − 1.28·61-s − 1.48·65-s − 1.46·67-s + 1.42·71-s − 1.17·73-s + 0.900·79-s + 1.31·83-s + 0.433·85-s − 1.05·89-s − 0.628·91-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 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−14.78512519676361, −13.80863876082357, −13.48868383260408, −13.25213303964815, −12.38824754484764, −12.05480637539909, −11.36763596201346, −11.13631938818252, −10.66134480730390, −9.878676256895812, −9.338439231436308, −8.938163364385856, −8.251920651122781, −7.907819586906680, −7.141943846489376, −6.890984725139767, −6.055058173129130, −5.622993272176501, −4.948129748577710, −4.248846052759145, −3.551680712394603, −3.384957075699430, −2.575524792081695, −1.507814403458567, −0.9582130678950920, 0, 0.9582130678950920, 1.507814403458567, 2.575524792081695, 3.384957075699430, 3.551680712394603, 4.248846052759145, 4.948129748577710, 5.622993272176501, 6.055058173129130, 6.890984725139767, 7.141943846489376, 7.907819586906680, 8.251920651122781, 8.938163364385856, 9.338439231436308, 9.878676256895812, 10.66134480730390, 11.13631938818252, 11.36763596201346, 12.05480637539909, 12.38824754484764, 13.25213303964815, 13.48868383260408, 13.80863876082357, 14.78512519676361

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