Properties

Label 2-187200-1.1-c1-0-132
Degree $2$
Conductor $187200$
Sign $1$
Analytic cond. $1494.79$
Root an. cond. $38.6626$
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 − 4·11-s + 13-s + 6·17-s + 4·23-s − 6·29-s + 8·31-s − 2·37-s − 10·41-s + 4·43-s − 8·47-s + 9·49-s + 2·53-s − 4·59-s − 14·61-s + 12·67-s − 8·71-s + 10·73-s − 16·77-s − 4·83-s − 10·89-s + 4·91-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  + 1.51·7-s − 1.20·11-s + 0.277·13-s + 1.45·17-s + 0.834·23-s − 1.11·29-s + 1.43·31-s − 0.328·37-s − 1.56·41-s + 0.609·43-s − 1.16·47-s + 9/7·49-s + 0.274·53-s − 0.520·59-s − 1.79·61-s + 1.46·67-s − 0.949·71-s + 1.17·73-s − 1.82·77-s − 0.439·83-s − 1.05·89-s + 0.419·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(187200\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(1494.79\)
Root analytic conductor: \(38.6626\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{187200} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 187200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.997473912\)
\(L(\frac12)\) \(\approx\) \(2.997473912\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 - T \)
good7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.18010700116537, −12.57070891645335, −12.20891584954403, −11.65294732915601, −11.21141403341226, −10.84602236395437, −10.34398069493100, −9.901592370863491, −9.393522531830230, −8.624366625952575, −8.288941281187847, −7.917752592162068, −7.477503335024156, −7.021267237850207, −6.235402070347608, −5.691200899005314, −5.112459051598173, −4.977446847444217, −4.340242996321007, −3.561076351074804, −3.071059537471554, −2.476972792651556, −1.686806487278740, −1.328166594766732, −0.5011023855468033, 0.5011023855468033, 1.328166594766732, 1.686806487278740, 2.476972792651556, 3.071059537471554, 3.561076351074804, 4.340242996321007, 4.977446847444217, 5.112459051598173, 5.691200899005314, 6.235402070347608, 7.021267237850207, 7.477503335024156, 7.917752592162068, 8.288941281187847, 8.624366625952575, 9.393522531830230, 9.901592370863491, 10.34398069493100, 10.84602236395437, 11.21141403341226, 11.65294732915601, 12.20891584954403, 12.57070891645335, 13.18010700116537

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