Properties

Label 2-121520-1.1-c1-0-12
Degree $2$
Conductor $121520$
Sign $1$
Analytic cond. $970.342$
Root an. cond. $31.1503$
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  + 2·3-s + 5-s + 9-s − 2·11-s + 2·15-s − 2·17-s − 4·19-s + 4·23-s + 25-s − 4·27-s − 4·29-s − 31-s − 4·33-s − 8·37-s − 6·41-s − 2·43-s + 45-s − 4·51-s + 8·53-s − 2·55-s − 8·57-s + 8·59-s − 4·67-s + 8·69-s − 6·73-s + 2·75-s + 4·79-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.447·5-s + 1/3·9-s − 0.603·11-s + 0.516·15-s − 0.485·17-s − 0.917·19-s + 0.834·23-s + 1/5·25-s − 0.769·27-s − 0.742·29-s − 0.179·31-s − 0.696·33-s − 1.31·37-s − 0.937·41-s − 0.304·43-s + 0.149·45-s − 0.560·51-s + 1.09·53-s − 0.269·55-s − 1.05·57-s + 1.04·59-s − 0.488·67-s + 0.963·69-s − 0.702·73-s + 0.230·75-s + 0.450·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 121520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(121520\)    =    \(2^{4} \cdot 5 \cdot 7^{2} \cdot 31\)
Sign: $1$
Analytic conductor: \(970.342\)
Root analytic conductor: \(31.1503\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 121520,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.526382683\)
\(L(\frac12)\) \(\approx\) \(2.526382683\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T \)
7 \( 1 \)
31 \( 1 + T \)
good3 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
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

−13.47034525067121, −13.31146449338066, −12.77256454229126, −12.24032590808159, −11.56567060309329, −11.10129807439571, −10.47400487750569, −10.19601703762750, −9.558883525027210, −8.947690080643831, −8.746947401971207, −8.300426303647206, −7.668612646151322, −7.155791313971879, −6.700622910779225, −6.042743051410995, −5.395669690874133, −5.001043440518800, −4.263185523516663, −3.626900042440264, −3.190898901929341, −2.482855679021779, −2.096485868278731, −1.522705863720687, −0.4142003543184528, 0.4142003543184528, 1.522705863720687, 2.096485868278731, 2.482855679021779, 3.190898901929341, 3.626900042440264, 4.263185523516663, 5.001043440518800, 5.395669690874133, 6.042743051410995, 6.700622910779225, 7.155791313971879, 7.668612646151322, 8.300426303647206, 8.746947401971207, 8.947690080643831, 9.558883525027210, 10.19601703762750, 10.47400487750569, 11.10129807439571, 11.56567060309329, 12.24032590808159, 12.77256454229126, 13.31146449338066, 13.47034525067121

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