Properties

Label 2-152592-1.1-c1-0-41
Degree $2$
Conductor $152592$
Sign $-1$
Analytic cond. $1218.45$
Root an. cond. $34.9063$
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 − 2·5-s − 2·7-s + 9-s − 11-s + 4·13-s − 2·15-s − 2·19-s − 2·21-s + 2·23-s − 25-s + 27-s − 2·29-s − 4·31-s − 33-s + 4·35-s − 6·37-s + 4·39-s − 6·41-s − 2·43-s − 2·45-s − 3·49-s − 12·53-s + 2·55-s − 2·57-s + 14·59-s − 6·61-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s − 0.755·7-s + 1/3·9-s − 0.301·11-s + 1.10·13-s − 0.516·15-s − 0.458·19-s − 0.436·21-s + 0.417·23-s − 1/5·25-s + 0.192·27-s − 0.371·29-s − 0.718·31-s − 0.174·33-s + 0.676·35-s − 0.986·37-s + 0.640·39-s − 0.937·41-s − 0.304·43-s − 0.298·45-s − 3/7·49-s − 1.64·53-s + 0.269·55-s − 0.264·57-s + 1.82·59-s − 0.768·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152592\)    =    \(2^{4} \cdot 3 \cdot 11 \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(1218.45\)
Root analytic conductor: \(34.9063\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 152592,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
11 \( 1 + T \)
17 \( 1 \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−13.44779554601698, −13.15265856231723, −12.67070158423406, −12.22748366392301, −11.64256474022097, −11.14875775570126, −10.75310568917392, −10.22836811748187, −9.641428327750274, −9.221324025209762, −8.606521474023265, −8.320034523168385, −7.810487357472680, −7.250498401837000, −6.762915873281567, −6.299218692995968, −5.696425893286019, −5.014556506600114, −4.481275241413333, −3.698330548041957, −3.503800729226088, −3.104804879209079, −2.150880777714445, −1.679317809856498, −0.7170164052490554, 0, 0.7170164052490554, 1.679317809856498, 2.150880777714445, 3.104804879209079, 3.503800729226088, 3.698330548041957, 4.481275241413333, 5.014556506600114, 5.696425893286019, 6.299218692995968, 6.762915873281567, 7.250498401837000, 7.810487357472680, 8.320034523168385, 8.606521474023265, 9.221324025209762, 9.641428327750274, 10.22836811748187, 10.75310568917392, 11.14875775570126, 11.64256474022097, 12.22748366392301, 12.67070158423406, 13.15265856231723, 13.44779554601698

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