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Properties

Label 2-129360-1.1-c1-0-130
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 − 4·13-s − 15-s − 6·17-s + 6·19-s + 25-s + 27-s − 2·29-s − 2·31-s + 33-s − 2·37-s − 4·39-s − 6·41-s + 4·43-s − 45-s + 2·47-s − 6·51-s − 2·53-s − 55-s + 6·57-s + 8·59-s + 8·61-s + 4·65-s + 12·67-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.301·11-s − 1.10·13-s − 0.258·15-s − 1.45·17-s + 1.37·19-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 0.359·31-s + 0.174·33-s − 0.328·37-s − 0.640·39-s − 0.937·41-s + 0.609·43-s − 0.149·45-s + 0.291·47-s − 0.840·51-s − 0.274·53-s − 0.134·55-s + 0.794·57-s + 1.04·59-s + 1.02·61-s + 0.496·65-s + 1.46·67-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: Trivial
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 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−13.76210521995988, −13.14802111574825, −12.96870391720108, −12.14444875789326, −11.92181394893105, −11.34740917938334, −10.91804955335296, −10.26407285916143, −9.736499035615579, −9.383364491855723, −8.831053914748173, −8.420574471193457, −7.780370841926360, −7.338052756267028, −6.900475964681446, −6.514381958986233, −5.527422850165967, −5.211006748994031, −4.521394378907086, −4.046836863968556, −3.484171990559912, −2.850864168100832, −2.298153586217056, −1.706852778279484, −0.8184270747177369, 0, 0.8184270747177369, 1.706852778279484, 2.298153586217056, 2.850864168100832, 3.484171990559912, 4.046836863968556, 4.521394378907086, 5.211006748994031, 5.527422850165967, 6.514381958986233, 6.900475964681446, 7.338052756267028, 7.780370841926360, 8.420574471193457, 8.831053914748173, 9.383364491855723, 9.736499035615579, 10.26407285916143, 10.91804955335296, 11.34740917938334, 11.92181394893105, 12.14444875789326, 12.96870391720108, 13.14802111574825, 13.76210521995988

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