Properties

Label 2-1200-1.1-c3-0-33
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 + 32·7-s + 9·9-s + 60·11-s + 34·13-s − 42·17-s + 76·19-s + 96·21-s + 27·27-s + 6·29-s + 232·31-s + 180·33-s − 134·37-s + 102·39-s + 234·41-s − 412·43-s − 360·47-s + 681·49-s − 126·51-s − 222·53-s + 228·57-s − 660·59-s − 490·61-s + 288·63-s + 812·67-s − 120·71-s − 746·73-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.72·7-s + 1/3·9-s + 1.64·11-s + 0.725·13-s − 0.599·17-s + 0.917·19-s + 0.997·21-s + 0.192·27-s + 0.0384·29-s + 1.34·31-s + 0.949·33-s − 0.595·37-s + 0.418·39-s + 0.891·41-s − 1.46·43-s − 1.11·47-s + 1.98·49-s − 0.345·51-s − 0.575·53-s + 0.529·57-s − 1.45·59-s − 1.02·61-s + 0.575·63-s + 1.48·67-s − 0.200·71-s − 1.19·73-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

\(L(2)\) \(\approx\) \(4.118714546\)
\(L(\frac12)\) \(\approx\) \(4.118714546\)
\(L(\frac{5}{2})\) 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 \( 1 - 32 T + p^{3} T^{2} \)
11 \( 1 - 60 T + p^{3} T^{2} \)
13 \( 1 - 34 T + p^{3} T^{2} \)
17 \( 1 + 42 T + p^{3} T^{2} \)
19 \( 1 - 4 p T + p^{3} T^{2} \)
23 \( 1 + p^{3} T^{2} \)
29 \( 1 - 6 T + p^{3} T^{2} \)
31 \( 1 - 232 T + p^{3} T^{2} \)
37 \( 1 + 134 T + p^{3} T^{2} \)
41 \( 1 - 234 T + p^{3} T^{2} \)
43 \( 1 + 412 T + p^{3} T^{2} \)
47 \( 1 + 360 T + p^{3} T^{2} \)
53 \( 1 + 222 T + p^{3} T^{2} \)
59 \( 1 + 660 T + p^{3} T^{2} \)
61 \( 1 + 490 T + p^{3} T^{2} \)
67 \( 1 - 812 T + p^{3} T^{2} \)
71 \( 1 + 120 T + p^{3} T^{2} \)
73 \( 1 + 746 T + p^{3} T^{2} \)
79 \( 1 + 152 T + p^{3} T^{2} \)
83 \( 1 + 804 T + p^{3} T^{2} \)
89 \( 1 + 678 T + p^{3} T^{2} \)
97 \( 1 + 2 p T + p^{3} T^{2} \)
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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

−9.164858327843259744665934240153, −8.545111329440150688332317992300, −7.927032409131370979175356898085, −6.97804788593288040156666683426, −6.10161346980439707981858285442, −4.86444372262751268927006658512, −4.23941902522525956850157665074, −3.19752820355494074183114053304, −1.76353162289035314921190208974, −1.18780418835045452296556171580, 1.18780418835045452296556171580, 1.76353162289035314921190208974, 3.19752820355494074183114053304, 4.23941902522525956850157665074, 4.86444372262751268927006658512, 6.10161346980439707981858285442, 6.97804788593288040156666683426, 7.927032409131370979175356898085, 8.545111329440150688332317992300, 9.164858327843259744665934240153

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