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Properties

Label 2-135240-1.1-c1-0-21
Degree $2$
Conductor $135240$
Sign $1$
Analytic cond. $1079.89$
Root an. cond. $32.8617$
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 − 6·13-s + 15-s + 4·17-s + 4·19-s + 23-s + 25-s + 27-s − 8·29-s − 4·31-s + 2·37-s − 6·39-s + 4·43-s + 45-s − 4·47-s + 4·51-s − 8·53-s + 4·57-s − 10·61-s − 6·65-s + 4·67-s + 69-s − 12·71-s − 12·73-s + 75-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.66·13-s + 0.258·15-s + 0.970·17-s + 0.917·19-s + 0.208·23-s + 1/5·25-s + 0.192·27-s − 1.48·29-s − 0.718·31-s + 0.328·37-s − 0.960·39-s + 0.609·43-s + 0.149·45-s − 0.583·47-s + 0.560·51-s − 1.09·53-s + 0.529·57-s − 1.28·61-s − 0.744·65-s + 0.488·67-s + 0.120·69-s − 1.42·71-s − 1.40·73-s + 0.115·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(135240\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(1079.89\)
Root analytic conductor: \(32.8617\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 135240,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.653293855\)
\(L(\frac12)\) \(\approx\) \(2.653293855\)
\(L(\frac{3}{2})\) 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 \( 1 \)
23 \( 1 - T \)
good11 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−13.43505016497765, −12.93808060803054, −12.62412044789481, −11.98758781123254, −11.69410287739665, −10.97193729263702, −10.45045624802048, −9.940309161064036, −9.483921450669780, −9.271561861093391, −8.703064197519381, −7.781408407481954, −7.601533093886746, −7.304363250618409, −6.552076459853089, −5.876494131784675, −5.435107379465616, −4.880635379221782, −4.432606141146639, −3.525956913904390, −3.195311271906656, −2.556885951902186, −1.933070941821900, −1.397134897086147, −0.4546848893183192, 0.4546848893183192, 1.397134897086147, 1.933070941821900, 2.556885951902186, 3.195311271906656, 3.525956913904390, 4.432606141146639, 4.880635379221782, 5.435107379465616, 5.876494131784675, 6.552076459853089, 7.304363250618409, 7.601533093886746, 7.781408407481954, 8.703064197519381, 9.271561861093391, 9.483921450669780, 9.940309161064036, 10.45045624802048, 10.97193729263702, 11.69410287739665, 11.98758781123254, 12.62412044789481, 12.93808060803054, 13.43505016497765

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