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Properties

Label 2-118800-1.1-c1-0-92
Degree $2$
Conductor $118800$
Sign $1$
Analytic cond. $948.622$
Root an. cond. $30.7997$
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  + 4·7-s + 11-s + 2·13-s + 3·17-s + 4·19-s − 6·23-s + 6·29-s − 5·31-s + 2·37-s − 3·41-s − 8·43-s + 6·47-s + 9·49-s − 12·53-s − 6·59-s − 4·61-s − 5·67-s + 12·71-s + 2·73-s + 4·77-s + 10·79-s + 9·83-s + 18·89-s + 8·91-s − 97-s + 101-s + 103-s + ⋯
L(s)  = 1  + 1.51·7-s + 0.301·11-s + 0.554·13-s + 0.727·17-s + 0.917·19-s − 1.25·23-s + 1.11·29-s − 0.898·31-s + 0.328·37-s − 0.468·41-s − 1.21·43-s + 0.875·47-s + 9/7·49-s − 1.64·53-s − 0.781·59-s − 0.512·61-s − 0.610·67-s + 1.42·71-s + 0.234·73-s + 0.455·77-s + 1.12·79-s + 0.987·83-s + 1.90·89-s + 0.838·91-s − 0.101·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 118800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 118800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(118800\)    =    \(2^{4} \cdot 3^{3} \cdot 5^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(948.622\)
Root analytic conductor: \(30.7997\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 118800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.778335631\)
\(L(\frac12)\) \(\approx\) \(3.778335631\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 - T \)
good7 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−13.74540224369684, −13.21641322451448, −12.38696326711440, −12.06335452519689, −11.72851527803853, −11.13850093430593, −10.76194389172191, −10.23225020667047, −9.661591095635382, −9.165470782382390, −8.553163364134243, −8.086407162849880, −7.729019757041997, −7.321352964515868, −6.403852433019532, −6.136165997410227, −5.373663060150516, −4.931139653013213, −4.539698401419140, −3.662598855632647, −3.420171203146060, −2.474051113354234, −1.762515033328033, −1.369962923142672, −0.6182185328294570, 0.6182185328294570, 1.369962923142672, 1.762515033328033, 2.474051113354234, 3.420171203146060, 3.662598855632647, 4.539698401419140, 4.931139653013213, 5.373663060150516, 6.136165997410227, 6.403852433019532, 7.321352964515868, 7.729019757041997, 8.086407162849880, 8.553163364134243, 9.165470782382390, 9.661591095635382, 10.23225020667047, 10.76194389172191, 11.13850093430593, 11.72851527803853, 12.06335452519689, 12.38696326711440, 13.21641322451448, 13.74540224369684

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