↑

Properties

Label 2-135240-1.1-c1-0-42
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s + 9-s + 3·11-s + 4·13-s + 15-s + 2·17-s + 4·19-s + 23-s + 25-s − 27-s + 4·31-s − 3·33-s + 3·37-s − 4·39-s + 43-s − 45-s − 7·47-s − 2·51-s + 2·53-s − 3·55-s − 4·57-s − 4·59-s + 14·61-s − 4·65-s − 5·67-s − 69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.904·11-s + 1.10·13-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 + 0.718·31-s − 0.522·33-s + 0.493·37-s − 0.640·39-s + 0.152·43-s − 0.149·45-s − 1.02·47-s − 0.280·51-s + 0.274·53-s − 0.404·55-s − 0.529·57-s − 0.520·59-s + 1.79·61-s − 0.496·65-s − 0.610·67-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

\(L(1)\) \(\approx\) \(2.824880833\)
\(L(\frac12)\) \(\approx\) \(2.824880833\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 + 11 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.48079505940046, −12.88586388853125, −12.38843194640021, −11.94747972871860, −11.50357820824177, −11.16534971977715, −10.70647101484655, −10.05018622640309, −9.590343380965109, −9.144965902118442, −8.548607335479962, −7.958252852053017, −7.681434623551134, −6.796784693187276, −6.574009755762682, −6.078746457791480, −5.330333596759501, −5.045040810917630, −4.235556306972798, −3.761266759956527, −3.372840946655519, −2.600554568241052, −1.702421160058060, −1.043734045348490, −0.6450635399871244, 0.6450635399871244, 1.043734045348490, 1.702421160058060, 2.600554568241052, 3.372840946655519, 3.761266759956527, 4.235556306972798, 5.045040810917630, 5.330333596759501, 6.078746457791480, 6.574009755762682, 6.796784693187276, 7.681434623551134, 7.958252852053017, 8.548607335479962, 9.144965902118442, 9.590343380965109, 10.05018622640309, 10.70647101484655, 11.16534971977715, 11.50357820824177, 11.94747972871860, 12.38843194640021, 12.88586388853125, 13.48079505940046

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