Properties

Label 2-30e2-1.1-c1-0-6
Degree $2$
Conductor $900$
Sign $-1$
Analytic cond. $7.18653$
Root an. cond. $2.68077$
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  − 7-s − 7·13-s − 7·19-s + 11·31-s − 10·37-s − 13·43-s − 6·49-s − 61-s + 11·67-s − 10·73-s − 4·79-s + 7·91-s − 19·97-s + 20·103-s + 17·109-s + ⋯
L(s)  = 1  − 0.377·7-s − 1.94·13-s − 1.60·19-s + 1.97·31-s − 1.64·37-s − 1.98·43-s − 6/7·49-s − 0.128·61-s + 1.34·67-s − 1.17·73-s − 0.450·79-s + 0.733·91-s − 1.92·97-s + 1.97·103-s + 1.62·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
good7 \( 1 + T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 7 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 11 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 13 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 19 T + p T^{2} \)
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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

−10.01185297753129750762994211064, −8.810318287138666656620561570522, −8.068267125649044096169699508794, −7.00418644170583896469357710759, −6.41391381288026324076920371033, −5.13958532772580839484877653078, −4.42116330171299264580891094162, −3.08601112889137016020769949109, −2.05095977025298622950154668732, 0, 2.05095977025298622950154668732, 3.08601112889137016020769949109, 4.42116330171299264580891094162, 5.13958532772580839484877653078, 6.41391381288026324076920371033, 7.00418644170583896469357710759, 8.068267125649044096169699508794, 8.810318287138666656620561570522, 10.01185297753129750762994211064

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