Properties

Label 2-1200-1.1-c3-0-51
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 $1$

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: \(1\)
Selberg data: \((2,\ 1200,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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

−8.694070151637486189019718805360, −8.332764124657891939209909511297, −7.43238125108599296220671629479, −6.56221507968919661873361177847, −5.44763920086435595426136820589, −4.70669791811104319580596608049, −3.47101906263729083197546502151, −2.68936112318262079800782738959, −1.53262138606236513808992856139, 0, 1.53262138606236513808992856139, 2.68936112318262079800782738959, 3.47101906263729083197546502151, 4.70669791811104319580596608049, 5.44763920086435595426136820589, 6.56221507968919661873361177847, 7.43238125108599296220671629479, 8.332764124657891939209909511297, 8.694070151637486189019718805360

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