Properties

Label 2-1200-1.1-c3-0-2
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 − 2·7-s + 9·9-s − 70·11-s − 54·13-s + 22·17-s − 24·19-s + 6·21-s − 100·23-s − 27·27-s + 216·29-s − 208·31-s + 210·33-s + 254·37-s + 162·39-s − 206·41-s + 292·43-s − 320·47-s − 339·49-s − 66·51-s + 402·53-s + 72·57-s + 370·59-s − 550·61-s − 18·63-s + 728·67-s + 300·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.107·7-s + 1/3·9-s − 1.91·11-s − 1.15·13-s + 0.313·17-s − 0.289·19-s + 0.0623·21-s − 0.906·23-s − 0.192·27-s + 1.38·29-s − 1.20·31-s + 1.10·33-s + 1.12·37-s + 0.665·39-s − 0.784·41-s + 1.03·43-s − 0.993·47-s − 0.988·49-s − 0.181·51-s + 1.04·53-s + 0.167·57-s + 0.816·59-s − 1.15·61-s − 0.0359·63-s + 1.32·67-s + 0.523·69-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

\(L(2)\) \(\approx\) \(0.7577443361\)
\(L(\frac12)\) \(\approx\) \(0.7577443361\)
\(L(\frac{5}{2})\) 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 \( 1 + 2 T + p^{3} T^{2} \)
11 \( 1 + 70 T + p^{3} T^{2} \)
13 \( 1 + 54 T + p^{3} T^{2} \)
17 \( 1 - 22 T + p^{3} T^{2} \)
19 \( 1 + 24 T + p^{3} T^{2} \)
23 \( 1 + 100 T + p^{3} T^{2} \)
29 \( 1 - 216 T + p^{3} T^{2} \)
31 \( 1 + 208 T + p^{3} T^{2} \)
37 \( 1 - 254 T + p^{3} T^{2} \)
41 \( 1 + 206 T + p^{3} T^{2} \)
43 \( 1 - 292 T + p^{3} T^{2} \)
47 \( 1 + 320 T + p^{3} T^{2} \)
53 \( 1 - 402 T + p^{3} T^{2} \)
59 \( 1 - 370 T + p^{3} T^{2} \)
61 \( 1 + 550 T + p^{3} T^{2} \)
67 \( 1 - 728 T + p^{3} T^{2} \)
71 \( 1 - 540 T + p^{3} T^{2} \)
73 \( 1 + 604 T + p^{3} T^{2} \)
79 \( 1 + 792 T + p^{3} T^{2} \)
83 \( 1 - 404 T + p^{3} T^{2} \)
89 \( 1 + 938 T + p^{3} T^{2} \)
97 \( 1 + 56 T + p^{3} T^{2} \)
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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

−9.693458728127956255151787252341, −8.360960469320114074023853196207, −7.72641374550593698295202583392, −6.94481541664549009755977916950, −5.86041282461895173777167325779, −5.17651087418967106764033408254, −4.40563105664808928337795122028, −3.00024550672871204800724702444, −2.10048576885252284601413899357, −0.43415509775667737184029057643, 0.43415509775667737184029057643, 2.10048576885252284601413899357, 3.00024550672871204800724702444, 4.40563105664808928337795122028, 5.17651087418967106764033408254, 5.86041282461895173777167325779, 6.94481541664549009755977916950, 7.72641374550593698295202583392, 8.360960469320114074023853196207, 9.693458728127956255151787252341

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