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Properties

Label 2-135240-1.1-c1-0-3
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 − 5·11-s − 4·13-s + 15-s − 2·17-s − 4·19-s + 23-s + 25-s − 27-s + 4·29-s + 4·31-s + 5·33-s + 7·37-s + 4·39-s + 4·41-s − 11·43-s − 45-s − 3·47-s + 2·51-s + 2·53-s + 5·55-s + 4·57-s − 6·61-s + 4·65-s + 15·67-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.50·11-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 + 0.742·29-s + 0.718·31-s + 0.870·33-s + 1.15·37-s + 0.640·39-s + 0.624·41-s − 1.67·43-s − 0.149·45-s − 0.437·47-s + 0.280·51-s + 0.274·53-s + 0.674·55-s + 0.529·57-s − 0.768·61-s + 0.496·65-s + 1.83·67-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

\(L(1)\) \(\approx\) \(0.4344183335\)
\(L(\frac12)\) \(\approx\) \(0.4344183335\)
\(L(\frac{3}{2})\) 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 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 15 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 + 5 T + p T^{2} \)
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

−13.32051035354434, −12.90151665930949, −12.51370663367187, −12.01434873523079, −11.52595182743851, −11.03531398827776, −10.59422472689607, −10.09083154294678, −9.810228522262976, −9.066187757014379, −8.426308928211953, −8.040521022331797, −7.560260117099263, −7.066325054587487, −6.444506779700615, −6.082378414948988, −5.269706604650269, −4.817565292291991, −4.608105674867098, −3.849746827810313, −3.054547510357312, −2.511792694247950, −2.071634188779231, −1.018633665332445, −0.2304832248442080, 0.2304832248442080, 1.018633665332445, 2.071634188779231, 2.511792694247950, 3.054547510357312, 3.849746827810313, 4.608105674867098, 4.817565292291991, 5.269706604650269, 6.082378414948988, 6.444506779700615, 7.066325054587487, 7.560260117099263, 8.040521022331797, 8.426308928211953, 9.066187757014379, 9.810228522262976, 10.09083154294678, 10.59422472689607, 11.03531398827776, 11.52595182743851, 12.01434873523079, 12.51370663367187, 12.90151665930949, 13.32051035354434

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