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Properties

Label 2-135240-1.1-c1-0-27
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 + 11-s + 2·13-s − 15-s + 2·17-s − 4·19-s + 23-s + 25-s − 27-s − 33-s − 37-s − 2·39-s + 2·41-s + 43-s + 45-s + 47-s − 2·51-s + 6·53-s + 55-s + 4·57-s − 6·59-s − 8·61-s + 2·65-s + 7·67-s − 69-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.301·11-s + 0.554·13-s − 0.258·15-s + 0.485·17-s − 0.917·19-s + 0.208·23-s + 1/5·25-s − 0.192·27-s − 0.174·33-s − 0.164·37-s − 0.320·39-s + 0.312·41-s + 0.152·43-s + 0.149·45-s + 0.145·47-s − 0.280·51-s + 0.824·53-s + 0.134·55-s + 0.529·57-s − 0.781·59-s − 1.02·61-s + 0.248·65-s + 0.855·67-s − 0.120·69-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

\(L(1)\) \(\approx\) \(2.159286766\)
\(L(\frac12)\) \(\approx\) \(2.159286766\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 - 11 T + p T^{2} \)
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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

−13.38036040547822, −12.88307475836687, −12.56246294005866, −11.97503259209672, −11.56031925451002, −10.93832884605319, −10.64462348777591, −10.15563188778732, −9.604585094893985, −9.083776304109415, −8.675014226731000, −8.027972308811886, −7.554049478070694, −6.858100524402168, −6.462802708366823, −6.013090206943609, −5.473108041615245, −5.006608218575099, −4.289827098526143, −3.891573392608388, −3.173795297500365, −2.498708436618498, −1.794867033258096, −1.215236445831262, −0.4851987631514857, 0.4851987631514857, 1.215236445831262, 1.794867033258096, 2.498708436618498, 3.173795297500365, 3.891573392608388, 4.289827098526143, 5.006608218575099, 5.473108041615245, 6.013090206943609, 6.462802708366823, 6.858100524402168, 7.554049478070694, 8.027972308811886, 8.675014226731000, 9.083776304109415, 9.604585094893985, 10.15563188778732, 10.64462348777591, 10.93832884605319, 11.56031925451002, 11.97503259209672, 12.56246294005866, 12.88307475836687, 13.38036040547822

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