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Properties

Label 2-135240-1.1-c1-0-38
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 + 2·13-s − 15-s + 6·17-s − 23-s + 25-s − 27-s + 6·29-s − 6·37-s − 2·39-s + 6·41-s + 45-s + 8·47-s − 6·51-s − 6·53-s + 12·59-s − 10·61-s + 2·65-s + 69-s + 6·73-s − 75-s + 8·79-s + 81-s + 6·85-s − 6·87-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.554·13-s − 0.258·15-s + 1.45·17-s − 0.208·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 0.986·37-s − 0.320·39-s + 0.937·41-s + 0.149·45-s + 1.16·47-s − 0.840·51-s − 0.824·53-s + 1.56·59-s − 1.28·61-s + 0.248·65-s + 0.120·69-s + 0.702·73-s − 0.115·75-s + 0.900·79-s + 1/9·81-s + 0.650·85-s − 0.643·87-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

\(L(1)\) \(\approx\) \(2.921176993\)
\(L(\frac12)\) \(\approx\) \(2.921176993\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−13.57439434393142, −12.79190348575797, −12.49786028441177, −12.01081022355512, −11.64756248550165, −10.83037749025980, −10.70580508311628, −10.05512897869969, −9.713092525885681, −9.148827140141589, −8.530357090770559, −8.105067291910260, −7.464825473182646, −7.028536237250139, −6.383020205476421, −5.912441522789033, −5.575009411512357, −4.941578360631889, −4.449256949487733, −3.695846696323623, −3.253323352688677, −2.517930589560191, −1.807114416002765, −1.110065187585405, −0.6086040517399773, 0.6086040517399773, 1.110065187585405, 1.807114416002765, 2.517930589560191, 3.253323352688677, 3.695846696323623, 4.449256949487733, 4.941578360631889, 5.575009411512357, 5.912441522789033, 6.383020205476421, 7.028536237250139, 7.464825473182646, 8.105067291910260, 8.530357090770559, 9.148827140141589, 9.713092525885681, 10.05512897869969, 10.70580508311628, 10.83037749025980, 11.64756248550165, 12.01081022355512, 12.49786028441177, 12.79190348575797, 13.57439434393142

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