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Properties

Label 2-135240-1.1-c1-0-14
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 − 6·11-s + 4·13-s − 15-s − 2·17-s − 4·19-s + 23-s + 25-s − 27-s + 2·29-s + 2·31-s + 6·33-s + 12·37-s − 4·39-s − 6·41-s − 4·43-s + 45-s + 4·47-s + 2·51-s + 12·53-s − 6·55-s + 4·57-s − 10·59-s + 10·61-s + 4·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.80·11-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 + 0.371·29-s + 0.359·31-s + 1.04·33-s + 1.97·37-s − 0.640·39-s − 0.937·41-s − 0.609·43-s + 0.149·45-s + 0.583·47-s + 0.280·51-s + 1.64·53-s − 0.809·55-s + 0.529·57-s − 1.30·59-s + 1.28·61-s + 0.496·65-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

\(L(1)\) \(\approx\) \(1.545087254\)
\(L(\frac12)\) \(\approx\) \(1.545087254\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 12 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−13.32953390784938, −13.13191121435597, −12.60748450274963, −11.97102625382897, −11.47064212438836, −10.88947955533868, −10.64527232195126, −10.18645026328138, −9.758161088855253, −9.005188958474642, −8.496521182381664, −8.185287127611059, −7.490838295797810, −7.026307906639688, −6.300930782356308, −6.009545947381771, −5.543904694857876, −4.842865915959549, −4.538448339775290, −3.821269182585894, −3.061045212501451, −2.489997233079057, −1.980215128622586, −1.113435449557878, −0.4182173336588154, 0.4182173336588154, 1.113435449557878, 1.980215128622586, 2.489997233079057, 3.061045212501451, 3.821269182585894, 4.538448339775290, 4.842865915959549, 5.543904694857876, 6.009545947381771, 6.300930782356308, 7.026307906639688, 7.490838295797810, 8.185287127611059, 8.496521182381664, 9.005188958474642, 9.758161088855253, 10.18645026328138, 10.64527232195126, 10.88947955533868, 11.47064212438836, 11.97102625382897, 12.60748450274963, 13.13191121435597, 13.32953390784938

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