Properties

Label 2-1200-1.1-c1-0-14
Degree $2$
Conductor $1200$
Sign $-1$
Analytic cond. $9.58204$
Root an. cond. $3.09548$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 7-s + 9-s − 6·11-s + 5·13-s − 6·17-s − 5·19-s − 21-s + 6·23-s − 27-s − 6·29-s + 31-s + 6·33-s + 2·37-s − 5·39-s + 43-s − 6·47-s − 6·49-s + 6·51-s − 12·53-s + 5·57-s + 6·59-s − 13·61-s + 63-s − 11·67-s − 6·69-s + 2·73-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.80·11-s + 1.38·13-s − 1.45·17-s − 1.14·19-s − 0.218·21-s + 1.25·23-s − 0.192·27-s − 1.11·29-s + 0.179·31-s + 1.04·33-s + 0.328·37-s − 0.800·39-s + 0.152·43-s − 0.875·47-s − 6/7·49-s + 0.840·51-s − 1.64·53-s + 0.662·57-s + 0.781·59-s − 1.66·61-s + 0.125·63-s − 1.34·67-s − 0.722·69-s + 0.234·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(9.58204\)
Root analytic conductor: \(3.09548\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1200} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1200,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
5 \( 1 \)
good7 \( 1 - T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−9.279692653561850448555307787769, −8.468231327893462435372408139803, −7.77221163592809854791946446876, −6.72617585632288603031226319311, −5.97534946548386281158286709106, −5.04294329753287530460269059022, −4.31968852301707049776817989287, −2.97413684260461707986207095036, −1.74800111788545442259234061107, 0, 1.74800111788545442259234061107, 2.97413684260461707986207095036, 4.31968852301707049776817989287, 5.04294329753287530460269059022, 5.97534946548386281158286709106, 6.72617585632288603031226319311, 7.77221163592809854791946446876, 8.468231327893462435372408139803, 9.279692653561850448555307787769

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