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Properties

Label 2-135240-1.1-c1-0-30
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 − 3·11-s + 15-s − 6·17-s + 8·19-s − 23-s + 25-s − 27-s + 4·29-s + 4·31-s + 3·33-s − 37-s − 4·41-s + 9·43-s − 45-s + 5·47-s + 6·51-s + 10·53-s + 3·55-s − 8·57-s + 14·61-s − 5·67-s + 69-s + 4·71-s − 8·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.904·11-s + 0.258·15-s − 1.45·17-s + 1.83·19-s − 0.208·23-s + 1/5·25-s − 0.192·27-s + 0.742·29-s + 0.718·31-s + 0.522·33-s − 0.164·37-s − 0.624·41-s + 1.37·43-s − 0.149·45-s + 0.729·47-s + 0.840·51-s + 1.37·53-s + 0.404·55-s − 1.05·57-s + 1.79·61-s − 0.610·67-s + 0.120·69-s + 0.474·71-s − 0.936·73-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

\(L(1)\) \(\approx\) \(1.713063752\)
\(L(\frac12)\) \(\approx\) \(1.713063752\)
\(L(\frac{3}{2})\) 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 \( 1 + 3 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 9 T + p T^{2} \)
47 \( 1 - 5 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−13.26976982004920, −13.12236168333472, −12.35538862074246, −11.91495669283294, −11.61183842585068, −11.09542706211473, −10.54595496303965, −10.21381335150976, −9.647479494705636, −9.060709583393773, −8.540095086384773, −8.053168432954513, −7.420292093290894, −7.100341140390106, −6.581304523672100, −5.862410175543307, −5.447096055101382, −4.912523147448815, −4.391505362357187, −3.882875456085186, −3.093932676829718, −2.597465543039942, −1.946898584303623, −0.9467532854553099, −0.5042958446516665, 0.5042958446516665, 0.9467532854553099, 1.946898584303623, 2.597465543039942, 3.093932676829718, 3.882875456085186, 4.391505362357187, 4.912523147448815, 5.447096055101382, 5.862410175543307, 6.581304523672100, 7.100341140390106, 7.420292093290894, 8.053168432954513, 8.540095086384773, 9.060709583393773, 9.647479494705636, 10.21381335150976, 10.54595496303965, 11.09542706211473, 11.61183842585068, 11.91495669283294, 12.35538862074246, 13.12236168333472, 13.26976982004920

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