Properties

Label 2-129360-1.1-c1-0-200
Degree $2$
Conductor $129360$
Sign $-1$
Analytic cond. $1032.94$
Root an. cond. $32.1394$
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  + 3-s − 5-s + 9-s + 11-s + 2·13-s − 15-s − 2·17-s + 4·19-s + 25-s + 27-s + 6·29-s + 33-s + 6·37-s + 2·39-s + 6·41-s + 4·43-s − 45-s − 2·51-s − 2·53-s − 55-s + 4·57-s + 4·59-s − 6·61-s − 2·65-s − 12·67-s − 10·73-s + 75-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.301·11-s + 0.554·13-s − 0.258·15-s − 0.485·17-s + 0.917·19-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.174·33-s + 0.986·37-s + 0.320·39-s + 0.937·41-s + 0.609·43-s − 0.149·45-s − 0.280·51-s − 0.274·53-s − 0.134·55-s + 0.529·57-s + 0.520·59-s − 0.768·61-s − 0.248·65-s − 1.46·67-s − 1.17·73-s + 0.115·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 129360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(129360\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(1032.94\)
Root analytic conductor: \(32.1394\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{129360} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 129360,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 \)
11 \( 1 - T \)
good13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−13.75024681847107, −13.27900907337966, −12.85099814210464, −12.26052822218042, −11.85093829765023, −11.26814000757180, −10.97471434865371, −10.23140646603373, −9.902893211523360, −9.161669018233398, −8.929806928751366, −8.376817713795854, −7.816591026614346, −7.418056237490746, −6.915372056391604, −6.223587388744542, −5.878762088156443, −5.081264602747668, −4.429184753267015, −4.129974720095185, −3.446334053752900, −2.836543533907597, −2.458957252006305, −1.396985581532317, −1.059888158080453, 0, 1.059888158080453, 1.396985581532317, 2.458957252006305, 2.836543533907597, 3.446334053752900, 4.129974720095185, 4.429184753267015, 5.081264602747668, 5.878762088156443, 6.223587388744542, 6.915372056391604, 7.418056237490746, 7.816591026614346, 8.376817713795854, 8.929806928751366, 9.161669018233398, 9.902893211523360, 10.23140646603373, 10.97471434865371, 11.26814000757180, 11.85093829765023, 12.26052822218042, 12.85099814210464, 13.27900907337966, 13.75024681847107

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