Properties

Label 2-135240-1.1-c1-0-51
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 − 4·11-s − 2·13-s − 15-s − 6·17-s − 2·19-s + 23-s + 25-s + 27-s − 2·29-s + 4·31-s − 4·33-s + 4·37-s − 2·39-s + 6·41-s − 12·43-s − 45-s + 12·47-s − 6·51-s − 2·53-s + 4·55-s − 2·57-s + 4·59-s − 10·61-s + 2·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.258·15-s − 1.45·17-s − 0.458·19-s + 0.208·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 0.718·31-s − 0.696·33-s + 0.657·37-s − 0.320·39-s + 0.937·41-s − 1.82·43-s − 0.149·45-s + 1.75·47-s − 0.840·51-s − 0.274·53-s + 0.539·55-s − 0.264·57-s + 0.520·59-s − 1.28·61-s + 0.248·65-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: \(1\)
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 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−13.74704703829698, −13.07404324400720, −12.87884861191309, −12.40752084542341, −11.68325881996603, −11.30155013860134, −10.76600315532370, −10.31580961148976, −9.856169410095159, −9.261948105301994, −8.660394944021883, −8.457061165417442, −7.742215243598447, −7.413985913216243, −6.926168140820338, −6.268990681324415, −5.754305229449042, −4.915843805075632, −4.640327443037716, −4.103416527091556, −3.404732372309611, −2.722609099063580, −2.391639384476414, −1.764842595453993, −0.7117951491920197, 0, 0.7117951491920197, 1.764842595453993, 2.391639384476414, 2.722609099063580, 3.404732372309611, 4.103416527091556, 4.640327443037716, 4.915843805075632, 5.754305229449042, 6.268990681324415, 6.926168140820338, 7.413985913216243, 7.742215243598447, 8.457061165417442, 8.660394944021883, 9.261948105301994, 9.856169410095159, 10.31580961148976, 10.76600315532370, 11.30155013860134, 11.68325881996603, 12.40752084542341, 12.87884861191309, 13.07404324400720, 13.74704703829698

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