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Properties

Label 2-129360-1.1-c1-0-103
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s + 9-s − 11-s + 4·13-s + 15-s + 7·17-s + 19-s + 3·23-s + 25-s + 27-s − 29-s + 10·31-s − 33-s + 8·37-s + 4·39-s − 10·41-s + 43-s + 45-s − 10·47-s + 7·51-s + 53-s − 55-s + 57-s − 3·59-s − 7·61-s + 4·65-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 + 1.69·17-s + 0.229·19-s + 0.625·23-s + 1/5·25-s + 0.192·27-s − 0.185·29-s + 1.79·31-s − 0.174·33-s + 1.31·37-s + 0.640·39-s − 1.56·41-s + 0.152·43-s + 0.149·45-s − 1.45·47-s + 0.980·51-s + 0.137·53-s − 0.134·55-s + 0.132·57-s − 0.390·59-s − 0.896·61-s + 0.496·65-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: \(0\)
Selberg data: \((2,\ 129360,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.130720437\)
\(L(\frac12)\) \(\approx\) \(5.130720437\)
\(L(\frac{3}{2})\) 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 \( 1 - 7 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 7 T + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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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

−13.48815964253792, −13.12088141950948, −12.68066494329231, −12.02141683635016, −11.61352947646844, −11.09070002096909, −10.37203047235639, −10.15515705920351, −9.561346393078823, −9.226180058419436, −8.432459410902368, −8.167931837624415, −7.766951828322010, −7.057501671329465, −6.492646907179276, −6.048993948949455, −5.459815911476270, −4.927883489139235, −4.352586647026592, −3.573246696322490, −3.144622791491730, −2.741038213627595, −1.834029551597404, −1.267316259733079, −0.7141644923786705, 0.7141644923786705, 1.267316259733079, 1.834029551597404, 2.741038213627595, 3.144622791491730, 3.573246696322490, 4.352586647026592, 4.927883489139235, 5.459815911476270, 6.048993948949455, 6.492646907179276, 7.057501671329465, 7.766951828322010, 8.167931837624415, 8.432459410902368, 9.226180058419436, 9.561346393078823, 10.15515705920351, 10.37203047235639, 11.09070002096909, 11.61352947646844, 12.02141683635016, 12.68066494329231, 13.12088141950948, 13.48815964253792

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