Properties

Label 2-60984-1.1-c1-0-34
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 + 5·13-s + 6·17-s − 8·19-s − 23-s − 25-s − 5·29-s + 5·31-s + 2·35-s − 8·37-s − 3·41-s − 43-s + 2·47-s + 49-s + 10·53-s − 3·59-s − 3·61-s − 10·65-s + 13·67-s − 15·71-s + 12·73-s + 8·79-s + 11·83-s − 12·85-s − 15·89-s − 5·91-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.377·7-s + 1.38·13-s + 1.45·17-s − 1.83·19-s − 0.208·23-s − 1/5·25-s − 0.928·29-s + 0.898·31-s + 0.338·35-s − 1.31·37-s − 0.468·41-s − 0.152·43-s + 0.291·47-s + 1/7·49-s + 1.37·53-s − 0.390·59-s − 0.384·61-s − 1.24·65-s + 1.58·67-s − 1.78·71-s + 1.40·73-s + 0.900·79-s + 1.20·83-s − 1.30·85-s − 1.58·89-s − 0.524·91-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 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 3 T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 11 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−14.54544937070908, −14.08278764232370, −13.41171581982244, −13.12106270210141, −12.40043741863083, −12.05542026674081, −11.61016003863871, −10.89446340408401, −10.57336221463865, −10.07033654132984, −9.376352477082846, −8.760367808623527, −8.256164739075303, −8.003043631074947, −7.231953800479037, −6.719200852694837, −6.088737466614753, −5.689664871663556, −4.918035278361191, −4.166721748322055, −3.634878773880974, −3.440538779456770, −2.425247575828370, −1.682258402872246, −0.8402601717546476, 0, 0.8402601717546476, 1.682258402872246, 2.425247575828370, 3.440538779456770, 3.634878773880974, 4.166721748322055, 4.918035278361191, 5.689664871663556, 6.088737466614753, 6.719200852694837, 7.231953800479037, 8.003043631074947, 8.256164739075303, 8.760367808623527, 9.376352477082846, 10.07033654132984, 10.57336221463865, 10.89446340408401, 11.61016003863871, 12.05542026674081, 12.40043741863083, 13.12106270210141, 13.41171581982244, 14.08278764232370, 14.54544937070908

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