Properties

Label 2-1200-1.1-c3-0-5
Degree $2$
Conductor $1200$
Sign $1$
Analytic cond. $70.8022$
Root an. cond. $8.41440$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 22.2·7-s + 9·9-s + 1.79·11-s + 58.2·13-s + 18.9·17-s − 104.·19-s + 66.6·21-s + 49.6·23-s − 27·27-s − 293.·29-s − 64.4·31-s − 5.37·33-s − 19.8·37-s − 174.·39-s − 165.·41-s + 247.·43-s − 384.·47-s + 150.·49-s − 56.9·51-s + 463.·53-s + 314.·57-s + 73.7·59-s − 137.·61-s − 199.·63-s − 173.·67-s − 148.·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.19·7-s + 0.333·9-s + 0.0490·11-s + 1.24·13-s + 0.270·17-s − 1.26·19-s + 0.692·21-s + 0.449·23-s − 0.192·27-s − 1.87·29-s − 0.373·31-s − 0.0283·33-s − 0.0883·37-s − 0.716·39-s − 0.630·41-s + 0.877·43-s − 1.19·47-s + 0.438·49-s − 0.156·51-s + 1.20·53-s + 0.730·57-s + 0.162·59-s − 0.288·61-s − 0.399·63-s − 0.317·67-s − 0.259·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(70.8022\)
Root analytic conductor: \(8.41440\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.098687614\)
\(L(\frac12)\) \(\approx\) \(1.098687614\)
\(L(\frac{5}{2})\) 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 \( 1 + 3T \)
5 \( 1 \)
good7 \( 1 + 22.2T + 343T^{2} \)
11 \( 1 - 1.79T + 1.33e3T^{2} \)
13 \( 1 - 58.2T + 2.19e3T^{2} \)
17 \( 1 - 18.9T + 4.91e3T^{2} \)
19 \( 1 + 104.T + 6.85e3T^{2} \)
23 \( 1 - 49.6T + 1.21e4T^{2} \)
29 \( 1 + 293.T + 2.43e4T^{2} \)
31 \( 1 + 64.4T + 2.97e4T^{2} \)
37 \( 1 + 19.8T + 5.06e4T^{2} \)
41 \( 1 + 165.T + 6.89e4T^{2} \)
43 \( 1 - 247.T + 7.95e4T^{2} \)
47 \( 1 + 384.T + 1.03e5T^{2} \)
53 \( 1 - 463.T + 1.48e5T^{2} \)
59 \( 1 - 73.7T + 2.05e5T^{2} \)
61 \( 1 + 137.T + 2.26e5T^{2} \)
67 \( 1 + 173.T + 3.00e5T^{2} \)
71 \( 1 - 594.T + 3.57e5T^{2} \)
73 \( 1 - 320.T + 3.89e5T^{2} \)
79 \( 1 - 770.T + 4.93e5T^{2} \)
83 \( 1 + 173.T + 5.71e5T^{2} \)
89 \( 1 - 1.01e3T + 7.04e5T^{2} \)
97 \( 1 - 384.T + 9.12e5T^{2} \)
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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

−9.365006049421048729686998040366, −8.724237432082986856781432388814, −7.62795954069138429535201246327, −6.64004705073133077799422442017, −6.15131461203116827960301539820, −5.29361751918668072950608277121, −4.01241031346673020480048410281, −3.35443132040750663178276035261, −1.91641532286767805795984862248, −0.54384344756174017117190352172, 0.54384344756174017117190352172, 1.91641532286767805795984862248, 3.35443132040750663178276035261, 4.01241031346673020480048410281, 5.29361751918668072950608277121, 6.15131461203116827960301539820, 6.64004705073133077799422442017, 7.62795954069138429535201246327, 8.724237432082986856781432388814, 9.365006049421048729686998040366

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