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Properties

Label 2-135240-1.1-c1-0-46
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s + 9-s − 3·11-s − 6·13-s − 15-s + 4·17-s − 8·19-s + 23-s + 25-s + 27-s − 2·29-s − 4·31-s − 3·33-s + 5·37-s − 6·39-s − 5·43-s − 45-s − 7·47-s + 4·51-s − 2·53-s + 3·55-s − 8·57-s + 2·61-s + 6·65-s + 67-s + 69-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.904·11-s − 1.66·13-s − 0.258·15-s + 0.970·17-s − 1.83·19-s + 0.208·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 0.718·31-s − 0.522·33-s + 0.821·37-s − 0.960·39-s − 0.762·43-s − 0.149·45-s − 1.02·47-s + 0.560·51-s − 0.274·53-s + 0.404·55-s − 1.05·57-s + 0.256·61-s + 0.744·65-s + 0.122·67-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: \(1\)
Selberg data: \((2,\ 135240,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 17 T + p T^{2} \)
97 \( 1 - 15 T + p T^{2} \)
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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

−13.66015862004207, −12.98713507513283, −12.81530344684844, −12.44267752767147, −11.71602467085088, −11.43144724405131, −10.49108437171471, −10.43479964072285, −9.881180805830443, −9.235084863568844, −8.895531867548405, −8.148053387896753, −7.750017004878842, −7.582599528961401, −6.823291878420789, −6.377785951848707, −5.617696402776472, −4.999209696615100, −4.677468231318067, −4.042978905157565, −3.341570750244238, −2.919282341151058, −2.147439391137267, −1.902539086646954, −0.6840048790536819, 0, 0.6840048790536819, 1.902539086646954, 2.147439391137267, 2.919282341151058, 3.341570750244238, 4.042978905157565, 4.677468231318067, 4.999209696615100, 5.617696402776472, 6.377785951848707, 6.823291878420789, 7.582599528961401, 7.750017004878842, 8.148053387896753, 8.895531867548405, 9.235084863568844, 9.881180805830443, 10.43479964072285, 10.49108437171471, 11.43144724405131, 11.71602467085088, 12.44267752767147, 12.81530344684844, 12.98713507513283, 13.66015862004207

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