Properties

Label 2-7200-1.1-c1-0-3
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  − 3·7-s − 4·11-s − 7·13-s − 4·17-s − 19-s + 8·23-s − 3·31-s + 2·37-s + 6·41-s − 11·43-s − 6·47-s + 2·49-s − 6·53-s + 6·59-s − 61-s − 15·67-s − 6·71-s + 2·73-s + 12·77-s − 8·79-s − 2·83-s + 16·89-s + 21·91-s + 13·97-s + 2·101-s − 16·103-s + 18·107-s + ⋯
L(s)  = 1  − 1.13·7-s − 1.20·11-s − 1.94·13-s − 0.970·17-s − 0.229·19-s + 1.66·23-s − 0.538·31-s + 0.328·37-s + 0.937·41-s − 1.67·43-s − 0.875·47-s + 2/7·49-s − 0.824·53-s + 0.781·59-s − 0.128·61-s − 1.83·67-s − 0.712·71-s + 0.234·73-s + 1.36·77-s − 0.900·79-s − 0.219·83-s + 1.69·89-s + 2.20·91-s + 1.31·97-s + 0.199·101-s − 1.57·103-s + 1.74·107-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

\(L(1)\) \(\approx\) \(0.5511855485\)
\(L(\frac12)\) \(\approx\) \(0.5511855485\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 7 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 15 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−7.73518458984781631479938463846, −7.21525048272757119527735815052, −6.65895029197279271200590164901, −5.84004587447743808171476362032, −4.91105587842042433939911556532, −4.64014047694129360989620976042, −3.28177997650748355599179591105, −2.80452156426397466716492169462, −2.01960379496755926227233066072, −0.34627339890746046357547877726, 0.34627339890746046357547877726, 2.01960379496755926227233066072, 2.80452156426397466716492169462, 3.28177997650748355599179591105, 4.64014047694129360989620976042, 4.91105587842042433939911556532, 5.84004587447743808171476362032, 6.65895029197279271200590164901, 7.21525048272757119527735815052, 7.73518458984781631479938463846

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