Properties

Label 2-300-1.1-c1-0-3
Degree $2$
Conductor $300$
Sign $-1$
Analytic cond. $2.39551$
Root an. cond. $1.54774$
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 − 4·7-s + 9-s − 4·11-s − 4·17-s + 4·21-s − 4·23-s − 27-s − 6·29-s + 4·31-s + 4·33-s + 8·37-s − 10·41-s − 4·43-s + 4·47-s + 9·49-s + 4·51-s + 12·53-s + 4·59-s + 2·61-s − 4·63-s + 4·67-s + 4·69-s + 8·73-s + 16·77-s − 12·79-s + 81-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.51·7-s + 1/3·9-s − 1.20·11-s − 0.970·17-s + 0.872·21-s − 0.834·23-s − 0.192·27-s − 1.11·29-s + 0.718·31-s + 0.696·33-s + 1.31·37-s − 1.56·41-s − 0.609·43-s + 0.583·47-s + 9/7·49-s + 0.560·51-s + 1.64·53-s + 0.520·59-s + 0.256·61-s − 0.503·63-s + 0.488·67-s + 0.481·69-s + 0.936·73-s + 1.82·77-s − 1.35·79-s + 1/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{300} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ -1)\)

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 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−11.25119555178140085424640669637, −10.24713640301064257194420457710, −9.671728604717533470308800943454, −8.425273316407858927756969056783, −7.18372888449204428119332901427, −6.30782253827108715004959607025, −5.36371477889983409574050795450, −3.96453757467671714692485940103, −2.56579130253629562111590352122, 0, 2.56579130253629562111590352122, 3.96453757467671714692485940103, 5.36371477889983409574050795450, 6.30782253827108715004959607025, 7.18372888449204428119332901427, 8.425273316407858927756969056783, 9.671728604717533470308800943454, 10.24713640301064257194420457710, 11.25119555178140085424640669637

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