Properties

Label 2-121680-1.1-c1-0-108
Degree $2$
Conductor $121680$
Sign $-1$
Analytic cond. $971.619$
Root an. cond. $31.1708$
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 + 4·7-s − 4·11-s − 6·17-s − 4·23-s + 25-s + 6·29-s − 8·31-s + 4·35-s + 2·37-s + 10·41-s + 4·43-s − 8·47-s + 9·49-s + 2·53-s − 4·55-s − 4·59-s + 14·61-s − 12·67-s + 8·71-s + 10·73-s − 16·77-s + 4·83-s − 6·85-s + 10·89-s + 2·97-s + 101-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.51·7-s − 1.20·11-s − 1.45·17-s − 0.834·23-s + 1/5·25-s + 1.11·29-s − 1.43·31-s + 0.676·35-s + 0.328·37-s + 1.56·41-s + 0.609·43-s − 1.16·47-s + 9/7·49-s + 0.274·53-s − 0.539·55-s − 0.520·59-s + 1.79·61-s − 1.46·67-s + 0.949·71-s + 1.17·73-s − 1.82·77-s + 0.439·83-s − 0.650·85-s + 1.05·89-s + 0.203·97-s + 0.0995·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 121680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(121680\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(971.619\)
Root analytic conductor: \(31.1708\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{121680} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 121680,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 - T \)
13 \( 1 \)
good7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−13.85481114716035, −13.19633570124460, −13.03517553696820, −12.33794609926214, −11.83582876760424, −11.22495387664946, −10.84873512106580, −10.63157950128221, −9.951739562587667, −9.334207456238760, −8.892946708350716, −8.286911711287551, −7.907566857202634, −7.551070510004102, −6.788645292597745, −6.328963090528082, −5.541192333836269, −5.305404838865891, −4.603688274738549, −4.327221705998957, −3.551478186213308, −2.546264579114250, −2.310589505717980, −1.728806057492208, −0.9151262092279977, 0, 0.9151262092279977, 1.728806057492208, 2.310589505717980, 2.546264579114250, 3.551478186213308, 4.327221705998957, 4.603688274738549, 5.305404838865891, 5.541192333836269, 6.328963090528082, 6.788645292597745, 7.551070510004102, 7.907566857202634, 8.286911711287551, 8.892946708350716, 9.334207456238760, 9.951739562587667, 10.63157950128221, 10.84873512106580, 11.22495387664946, 11.83582876760424, 12.33794609926214, 13.03517553696820, 13.19633570124460, 13.85481114716035

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