Properties

Label 2-187200-1.1-c1-0-118
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  − 7-s + 5·11-s − 13-s − 3·17-s + 4·23-s + 9·29-s − 7·31-s + 8·37-s + 2·41-s − 7·47-s − 6·49-s − 3·53-s − 9·59-s + 15·61-s + 7·67-s − 4·73-s − 5·77-s − 8·79-s + 7·83-s + 12·89-s + 91-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 0.377·7-s + 1.50·11-s − 0.277·13-s − 0.727·17-s + 0.834·23-s + 1.67·29-s − 1.25·31-s + 1.31·37-s + 0.312·41-s − 1.02·47-s − 6/7·49-s − 0.412·53-s − 1.17·59-s + 1.92·61-s + 0.855·67-s − 0.468·73-s − 0.569·77-s − 0.900·79-s + 0.768·83-s + 1.27·89-s + 0.104·91-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 187200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.442718022\)
\(L(\frac12)\) \(\approx\) \(2.442718022\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 - 15 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−13.00783238605268, −12.72176765647274, −12.21568579764640, −11.67277205477118, −11.19187167313721, −11.00441919806237, −10.17491671149447, −9.762832594880104, −9.321550139561865, −8.930365362194513, −8.441066336749147, −7.882529696213651, −7.233426325211327, −6.760781906343315, −6.390260346052540, −6.038490739164474, −5.119975528806237, −4.819786900984646, −4.094254401562348, −3.769702413617448, −3.007533618832024, −2.579024641656187, −1.748433577173964, −1.193937966976724, −0.4735361525703584, 0.4735361525703584, 1.193937966976724, 1.748433577173964, 2.579024641656187, 3.007533618832024, 3.769702413617448, 4.094254401562348, 4.819786900984646, 5.119975528806237, 6.038490739164474, 6.390260346052540, 6.760781906343315, 7.233426325211327, 7.882529696213651, 8.441066336749147, 8.930365362194513, 9.321550139561865, 9.762832594880104, 10.17491671149447, 11.00441919806237, 11.19187167313721, 11.67277205477118, 12.21568579764640, 12.72176765647274, 13.00783238605268

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