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Properties

Label 2-129360-1.1-c1-0-106
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 − 13-s + 15-s − 2·19-s + 9·23-s + 25-s − 27-s − 9·29-s − 8·31-s − 33-s + 2·37-s + 39-s + 3·41-s + 43-s − 45-s − 9·47-s + 9·53-s − 55-s + 2·57-s − 12·59-s − 10·61-s + 65-s + 4·67-s − 9·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.301·11-s − 0.277·13-s + 0.258·15-s − 0.458·19-s + 1.87·23-s + 1/5·25-s − 0.192·27-s − 1.67·29-s − 1.43·31-s − 0.174·33-s + 0.328·37-s + 0.160·39-s + 0.468·41-s + 0.152·43-s − 0.149·45-s − 1.31·47-s + 1.23·53-s − 0.134·55-s + 0.264·57-s − 1.56·59-s − 1.28·61-s + 0.124·65-s + 0.488·67-s − 1.08·69-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 \( 1 + T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 9 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
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

−13.58228594358913, −13.10458683347545, −12.75180396397433, −12.33890355159114, −11.77337400924283, −11.13866157997573, −11.00918181818343, −10.61081198870670, −9.727663355378745, −9.392620626548646, −8.945710513239769, −8.373872837359636, −7.680753280615828, −7.230105607209640, −6.942425478130578, −6.248580540651332, −5.646874366128861, −5.280380975450115, −4.537667838438320, −4.243132604608045, −3.394123280608540, −3.085312170732294, −2.091134374989174, −1.562425734679257, −0.7251315502423034, 0, 0.7251315502423034, 1.562425734679257, 2.091134374989174, 3.085312170732294, 3.394123280608540, 4.243132604608045, 4.537667838438320, 5.280380975450115, 5.646874366128861, 6.248580540651332, 6.942425478130578, 7.230105607209640, 7.680753280615828, 8.373872837359636, 8.945710513239769, 9.392620626548646, 9.727663355378745, 10.61081198870670, 11.00918181818343, 11.13866157997573, 11.77337400924283, 12.33890355159114, 12.75180396397433, 13.10458683347545, 13.58228594358913

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