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Properties

Label 2-135240-1.1-c1-0-13
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 + 4·11-s − 13-s − 15-s − 4·17-s + 5·19-s − 23-s + 25-s − 27-s − 9·29-s − 31-s − 4·33-s + 4·37-s + 39-s − 2·41-s + 45-s − 4·47-s + 4·51-s + 3·53-s + 4·55-s − 5·57-s − 7·59-s − 11·61-s − 65-s − 4·67-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s + 1.20·11-s − 0.277·13-s − 0.258·15-s − 0.970·17-s + 1.14·19-s − 0.208·23-s + 1/5·25-s − 0.192·27-s − 1.67·29-s − 0.179·31-s − 0.696·33-s + 0.657·37-s + 0.160·39-s − 0.312·41-s + 0.149·45-s − 0.583·47-s + 0.560·51-s + 0.412·53-s + 0.539·55-s − 0.662·57-s − 0.911·59-s − 1.40·61-s − 0.124·65-s − 0.488·67-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

\(L(1)\) \(\approx\) \(1.730197631\)
\(L(\frac12)\) \(\approx\) \(1.730197631\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + 7 T + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−13.38044675203016, −13.11732598275078, −12.32662259560428, −11.94682532173586, −11.64371262774455, −10.94070501234506, −10.77929075731888, −10.00753196167185, −9.410920687883758, −9.295429185200421, −8.783616335528706, −7.900678896969954, −7.518512545561749, −6.935049995717106, −6.471464844688982, −5.974182228380497, −5.552310907132426, −4.838523807394752, −4.475866753123855, −3.734457749851763, −3.301341355449084, −2.430484873151231, −1.765570441695682, −1.300184697402591, −0.4166061108357860, 0.4166061108357860, 1.300184697402591, 1.765570441695682, 2.430484873151231, 3.301341355449084, 3.734457749851763, 4.475866753123855, 4.838523807394752, 5.552310907132426, 5.974182228380497, 6.471464844688982, 6.935049995717106, 7.518512545561749, 7.900678896969954, 8.783616335528706, 9.295429185200421, 9.410920687883758, 10.00753196167185, 10.77929075731888, 10.94070501234506, 11.64371262774455, 11.94682532173586, 12.32662259560428, 13.11732598275078, 13.38044675203016

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