Properties

Label 2-1200-1.1-c3-0-55
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 + 20·7-s + 9·9-s − 16·11-s − 58·13-s − 38·17-s − 4·19-s + 60·21-s − 80·23-s + 27·27-s + 82·29-s + 8·31-s − 48·33-s − 426·37-s − 174·39-s − 246·41-s − 524·43-s − 464·47-s + 57·49-s − 114·51-s + 702·53-s − 12·57-s + 592·59-s + 574·61-s + 180·63-s − 172·67-s − 240·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.07·7-s + 1/3·9-s − 0.438·11-s − 1.23·13-s − 0.542·17-s − 0.0482·19-s + 0.623·21-s − 0.725·23-s + 0.192·27-s + 0.525·29-s + 0.0463·31-s − 0.253·33-s − 1.89·37-s − 0.714·39-s − 0.937·41-s − 1.85·43-s − 1.44·47-s + 0.166·49-s − 0.313·51-s + 1.81·53-s − 0.0278·57-s + 1.30·59-s + 1.20·61-s + 0.359·63-s − 0.313·67-s − 0.418·69-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 \( 1 - 20 T + p^{3} T^{2} \)
11 \( 1 + 16 T + p^{3} T^{2} \)
13 \( 1 + 58 T + p^{3} T^{2} \)
17 \( 1 + 38 T + p^{3} T^{2} \)
19 \( 1 + 4 T + p^{3} T^{2} \)
23 \( 1 + 80 T + p^{3} T^{2} \)
29 \( 1 - 82 T + p^{3} T^{2} \)
31 \( 1 - 8 T + p^{3} T^{2} \)
37 \( 1 + 426 T + p^{3} T^{2} \)
41 \( 1 + 6 p T + p^{3} T^{2} \)
43 \( 1 + 524 T + p^{3} T^{2} \)
47 \( 1 + 464 T + p^{3} T^{2} \)
53 \( 1 - 702 T + p^{3} T^{2} \)
59 \( 1 - 592 T + p^{3} T^{2} \)
61 \( 1 - 574 T + p^{3} T^{2} \)
67 \( 1 + 172 T + p^{3} T^{2} \)
71 \( 1 + 768 T + p^{3} T^{2} \)
73 \( 1 - 558 T + p^{3} T^{2} \)
79 \( 1 + 408 T + p^{3} T^{2} \)
83 \( 1 - 164 T + p^{3} T^{2} \)
89 \( 1 + 510 T + p^{3} T^{2} \)
97 \( 1 + 514 T + p^{3} T^{2} \)
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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

−8.661866669224269565481907260401, −8.325757662671788554122809905613, −7.38975164334645866079613442341, −6.69880147011743094152562042439, −5.25368741174324151897871037653, −4.78028609485695993390311496027, −3.64134507814836246018594730620, −2.44279926683918990574622020367, −1.66931161354033887472717155188, 0, 1.66931161354033887472717155188, 2.44279926683918990574622020367, 3.64134507814836246018594730620, 4.78028609485695993390311496027, 5.25368741174324151897871037653, 6.69880147011743094152562042439, 7.38975164334645866079613442341, 8.325757662671788554122809905613, 8.661866669224269565481907260401

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