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Properties

Label 2-7200-1.1-c1-0-29
Degree $2$
Conductor $7200$
Sign $1$
Analytic cond. $57.4922$
Root an. cond. $7.58236$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·13-s + 2·17-s + 4·29-s − 12·37-s + 8·41-s − 7·49-s + 14·53-s + 10·61-s + 16·73-s − 16·89-s − 8·97-s − 20·101-s − 6·109-s + 14·113-s + ⋯
L(s)  = 1  + 1.10·13-s + 0.485·17-s + 0.742·29-s − 1.97·37-s + 1.24·41-s − 49-s + 1.92·53-s + 1.28·61-s + 1.87·73-s − 1.69·89-s − 0.812·97-s − 1.99·101-s − 0.574·109-s + 1.31·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7200\)    =    \(2^{5} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(57.4922\)
Root analytic conductor: \(7.58236\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.175723845\)
\(L(\frac12)\) \(\approx\) \(2.175723845\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 12 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
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

−8.134981812132933062884367318080, −7.08564802969729138082623550983, −6.64106416224541718158337983471, −5.73177960357969520686867364811, −5.25380619985180969922896787586, −4.22006931432695016863165642875, −3.61240385453232533909280078955, −2.77238840879191056562050064581, −1.72949605868864546159014839141, −0.77395697445145218023139805994, 0.77395697445145218023139805994, 1.72949605868864546159014839141, 2.77238840879191056562050064581, 3.61240385453232533909280078955, 4.22006931432695016863165642875, 5.25380619985180969922896787586, 5.73177960357969520686867364811, 6.64106416224541718158337983471, 7.08564802969729138082623550983, 8.134981812132933062884367318080

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