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Properties

Label 2-135240-1.1-c1-0-15
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 + 2·11-s − 4·13-s − 15-s + 17-s − 2·19-s − 23-s + 25-s + 27-s + 2·29-s + 2·33-s − 6·37-s − 4·39-s + 9·41-s − 13·43-s − 45-s + 8·47-s + 51-s + 3·53-s − 2·55-s − 2·57-s − 5·59-s − 2·61-s + 4·65-s + 13·67-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.603·11-s − 1.10·13-s − 0.258·15-s + 0.242·17-s − 0.458·19-s − 0.208·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.348·33-s − 0.986·37-s − 0.640·39-s + 1.40·41-s − 1.98·43-s − 0.149·45-s + 1.16·47-s + 0.140·51-s + 0.412·53-s − 0.269·55-s − 0.264·57-s − 0.650·59-s − 0.256·61-s + 0.496·65-s + 1.58·67-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

\(L(1)\) \(\approx\) \(2.052853470\)
\(L(\frac12)\) \(\approx\) \(2.052853470\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + 13 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 + 13 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−13.42104008412103, −13.01745158458092, −12.38401380259769, −12.01776596115701, −11.74329169095537, −11.02394091206513, −10.35767800989949, −10.21015609479749, −9.425119816213279, −9.105694032301749, −8.603658436471592, −7.997752225892111, −7.655299199450050, −7.018538448591297, −6.718495017687787, −6.005215560626970, −5.390805016355301, −4.711052732969954, −4.361560070801012, −3.689474861163252, −3.222236221268968, −2.538289004933283, −2.003013133990856, −1.274508825565234, −0.4184575519738328, 0.4184575519738328, 1.274508825565234, 2.003013133990856, 2.538289004933283, 3.222236221268968, 3.689474861163252, 4.361560070801012, 4.711052732969954, 5.390805016355301, 6.005215560626970, 6.718495017687787, 7.018538448591297, 7.655299199450050, 7.997752225892111, 8.603658436471592, 9.105694032301749, 9.425119816213279, 10.21015609479749, 10.35767800989949, 11.02394091206513, 11.74329169095537, 12.01776596115701, 12.38401380259769, 13.01745158458092, 13.42104008412103

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