Properties

Label 2-1200-1.1-c3-0-9
Degree $2$
Conductor $1200$
Sign $1$
Analytic cond. $70.8022$
Root an. cond. $8.41440$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 16·7-s + 9·9-s + 28·11-s + 26·13-s + 62·17-s + 68·19-s + 48·21-s − 208·23-s − 27·27-s − 58·29-s − 160·31-s − 84·33-s − 270·37-s − 78·39-s + 282·41-s + 76·43-s − 280·47-s − 87·49-s − 186·51-s + 210·53-s − 204·57-s − 196·59-s + 742·61-s − 144·63-s + 836·67-s + 624·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.863·7-s + 1/3·9-s + 0.767·11-s + 0.554·13-s + 0.884·17-s + 0.821·19-s + 0.498·21-s − 1.88·23-s − 0.192·27-s − 0.371·29-s − 0.926·31-s − 0.443·33-s − 1.19·37-s − 0.320·39-s + 1.07·41-s + 0.269·43-s − 0.868·47-s − 0.253·49-s − 0.510·51-s + 0.544·53-s − 0.474·57-s − 0.432·59-s + 1.55·61-s − 0.287·63-s + 1.52·67-s + 1.08·69-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.430264671\)
\(L(\frac12)\) \(\approx\) \(1.430264671\)
\(L(\frac{5}{2})\) 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 \( 1 + p T \)
5 \( 1 \)
good7 \( 1 + 16 T + p^{3} T^{2} \)
11 \( 1 - 28 T + p^{3} T^{2} \)
13 \( 1 - 2 p T + p^{3} T^{2} \)
17 \( 1 - 62 T + p^{3} T^{2} \)
19 \( 1 - 68 T + p^{3} T^{2} \)
23 \( 1 + 208 T + p^{3} T^{2} \)
29 \( 1 + 2 p T + p^{3} T^{2} \)
31 \( 1 + 160 T + p^{3} T^{2} \)
37 \( 1 + 270 T + p^{3} T^{2} \)
41 \( 1 - 282 T + p^{3} T^{2} \)
43 \( 1 - 76 T + p^{3} T^{2} \)
47 \( 1 + 280 T + p^{3} T^{2} \)
53 \( 1 - 210 T + p^{3} T^{2} \)
59 \( 1 + 196 T + p^{3} T^{2} \)
61 \( 1 - 742 T + p^{3} T^{2} \)
67 \( 1 - 836 T + p^{3} T^{2} \)
71 \( 1 - 504 T + p^{3} T^{2} \)
73 \( 1 - 1062 T + p^{3} T^{2} \)
79 \( 1 + 768 T + p^{3} T^{2} \)
83 \( 1 + 1052 T + p^{3} T^{2} \)
89 \( 1 + 726 T + p^{3} T^{2} \)
97 \( 1 - 1406 T + p^{3} T^{2} \)
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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

−9.673805229660027357479661746545, −8.599008390408547256070560935664, −7.62729985796345273819228240988, −6.76600523945491710353833588611, −6.00142233326510178495054759928, −5.35223090894002274631691178930, −3.98039778451010378890800844263, −3.40402028316168536805962393537, −1.85186757735223223678678536801, −0.63692868567686991242973265837, 0.63692868567686991242973265837, 1.85186757735223223678678536801, 3.40402028316168536805962393537, 3.98039778451010378890800844263, 5.35223090894002274631691178930, 6.00142233326510178495054759928, 6.76600523945491710353833588611, 7.62729985796345273819228240988, 8.599008390408547256070560935664, 9.673805229660027357479661746545

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