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Properties

Label 2-129360-1.1-c1-0-174
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 − 5·13-s + 15-s + 2·19-s + 3·23-s + 25-s + 27-s + 3·29-s + 8·31-s − 33-s + 8·37-s − 5·39-s − 3·41-s − 5·43-s + 45-s + 3·47-s − 9·53-s − 55-s + 2·57-s − 2·61-s − 5·65-s − 14·67-s + 3·69-s − 12·71-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.301·11-s − 1.38·13-s + 0.258·15-s + 0.458·19-s + 0.625·23-s + 1/5·25-s + 0.192·27-s + 0.557·29-s + 1.43·31-s − 0.174·33-s + 1.31·37-s − 0.800·39-s − 0.468·41-s − 0.762·43-s + 0.149·45-s + 0.437·47-s − 1.23·53-s − 0.134·55-s + 0.264·57-s − 0.256·61-s − 0.620·65-s − 1.71·67-s + 0.361·69-s − 1.42·71-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 \( 1 + 5 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−13.67008736243575, −13.36295610716303, −12.83451426573214, −12.31164892772877, −11.88202908337385, −11.42138539236503, −10.65258018242365, −10.27548517810044, −9.781289044680874, −9.453684278895167, −8.905563929051238, −8.348259824513653, −7.734061791810249, −7.517339068911325, −6.746383227356568, −6.415226549956316, −5.686619902475591, −5.070451439728456, −4.653783765705431, −4.197526465324909, −3.190564051396807, −2.852347261200019, −2.412594310047798, −1.617763746107591, −0.9670726358137830, 0, 0.9670726358137830, 1.617763746107591, 2.412594310047798, 2.852347261200019, 3.190564051396807, 4.197526465324909, 4.653783765705431, 5.070451439728456, 5.686619902475591, 6.415226549956316, 6.746383227356568, 7.517339068911325, 7.734061791810249, 8.348259824513653, 8.905563929051238, 9.453684278895167, 9.781289044680874, 10.27548517810044, 10.65258018242365, 11.42138539236503, 11.88202908337385, 12.31164892772877, 12.83451426573214, 13.36295610716303, 13.67008736243575

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