Properties

Label 2-187200-1.1-c1-0-109
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  + 2·7-s − 2·11-s − 13-s + 2·17-s − 4·19-s + 4·29-s − 8·31-s + 6·37-s + 6·41-s − 4·43-s + 8·47-s − 3·49-s − 2·53-s − 10·59-s + 14·61-s + 16·67-s − 4·71-s − 8·73-s − 4·77-s + 8·79-s + 12·83-s − 6·89-s − 2·91-s − 12·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  + 0.755·7-s − 0.603·11-s − 0.277·13-s + 0.485·17-s − 0.917·19-s + 0.742·29-s − 1.43·31-s + 0.986·37-s + 0.937·41-s − 0.609·43-s + 1.16·47-s − 3/7·49-s − 0.274·53-s − 1.30·59-s + 1.79·61-s + 1.95·67-s − 0.474·71-s − 0.936·73-s − 0.455·77-s + 0.900·79-s + 1.31·83-s − 0.635·89-s − 0.209·91-s − 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-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

\(L(1)\) \(\approx\) \(2.356224175\)
\(L(\frac12)\) \(\approx\) \(2.356224175\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−12.98984598122707, −12.66424145475778, −12.31903082198050, −11.60461360728041, −11.24501352944614, −10.78128850683479, −10.39925137306480, −9.822948699157504, −9.347568564106178, −8.815220380173942, −8.231599686335149, −7.932122814025349, −7.441451767534238, −6.895624421387608, −6.330058617791344, −5.697808128778615, −5.341193205085550, −4.699241898765632, −4.330036806723996, −3.662761199184319, −3.051060523641975, −2.313475001461312, −1.999870216765581, −1.145107646657868, −0.4626002771603013, 0.4626002771603013, 1.145107646657868, 1.999870216765581, 2.313475001461312, 3.051060523641975, 3.662761199184319, 4.330036806723996, 4.699241898765632, 5.341193205085550, 5.697808128778615, 6.330058617791344, 6.895624421387608, 7.441451767534238, 7.932122814025349, 8.231599686335149, 8.815220380173942, 9.347568564106178, 9.822948699157504, 10.39925137306480, 10.78128850683479, 11.24501352944614, 11.60461360728041, 12.31903082198050, 12.66424145475778, 12.98984598122707

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