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Properties

Label 2-7200-1.1-c1-0-71
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·13-s + 8·17-s − 10·29-s − 12·37-s + 10·41-s − 7·49-s − 4·53-s + 10·61-s − 16·73-s + 10·89-s + 8·97-s + 2·101-s + 6·109-s − 16·113-s + ⋯
L(s)  = 1  − 1.10·13-s + 1.94·17-s − 1.85·29-s − 1.97·37-s + 1.56·41-s − 49-s − 0.549·53-s + 1.28·61-s − 1.87·73-s + 1.05·89-s + 0.812·97-s + 0.199·101-s + 0.574·109-s − 1.50·113-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: \(1\)
Selberg data: \((2,\ 7200,\ (\ :1/2),\ -1)\)

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 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 12 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
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 \( 1 - 10 T + p T^{2} \)
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

−7.51583516996385738039348254695, −7.10943363392382682125900106744, −6.05564796932311248390334150513, −5.43384591050843804641968621585, −4.88155016460207921995716576085, −3.82656131664777441755112073485, −3.22828886277885072977164990743, −2.25575881305648415025449876709, −1.31294899366076819859044923313, 0, 1.31294899366076819859044923313, 2.25575881305648415025449876709, 3.22828886277885072977164990743, 3.82656131664777441755112073485, 4.88155016460207921995716576085, 5.43384591050843804641968621585, 6.05564796932311248390334150513, 7.10943363392382682125900106744, 7.51583516996385738039348254695

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