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Properties

Label 2-7200-1.1-c1-0-84
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  + 3·7-s − 5·13-s − 5·19-s + 4·23-s − 4·29-s + 5·31-s − 10·37-s + 10·41-s − 43-s − 2·47-s + 2·49-s + 10·53-s − 10·59-s − 5·61-s + 3·67-s − 10·71-s − 10·73-s − 14·83-s − 16·89-s − 15·91-s − 5·97-s − 2·101-s − 16·103-s + 18·107-s + 5·109-s + 20·113-s + ⋯
L(s)  = 1  + 1.13·7-s − 1.38·13-s − 1.14·19-s + 0.834·23-s − 0.742·29-s + 0.898·31-s − 1.64·37-s + 1.56·41-s − 0.152·43-s − 0.291·47-s + 2/7·49-s + 1.37·53-s − 1.30·59-s − 0.640·61-s + 0.366·67-s − 1.18·71-s − 1.17·73-s − 1.53·83-s − 1.69·89-s − 1.57·91-s − 0.507·97-s − 0.199·101-s − 1.57·103-s + 1.74·107-s + 0.478·109-s + 1.88·113-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 \( 1 - 3 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−7.39404889454098030005239178255, −7.17488070082270852870041339846, −6.12205031663590669812429780346, −5.36206981807725594754969018860, −4.67359555617123472371661405063, −4.21836205659018076205085611258, −3.00440120076729956250344651914, −2.23162395018897564807692177746, −1.39823036021708305216897958246, 0, 1.39823036021708305216897958246, 2.23162395018897564807692177746, 3.00440120076729956250344651914, 4.21836205659018076205085611258, 4.67359555617123472371661405063, 5.36206981807725594754969018860, 6.12205031663590669812429780346, 7.17488070082270852870041339846, 7.39404889454098030005239178255

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