Properties

Label 2-187200-1.1-c1-0-121
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  + 6·11-s − 13-s + 2·17-s + 4·23-s − 6·29-s − 4·31-s − 2·37-s + 4·43-s + 10·47-s − 7·49-s + 10·53-s − 6·59-s + 6·61-s − 12·67-s − 2·71-s − 6·73-s − 16·79-s − 6·83-s − 4·89-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 1.80·11-s − 0.277·13-s + 0.485·17-s + 0.834·23-s − 1.11·29-s − 0.718·31-s − 0.328·37-s + 0.609·43-s + 1.45·47-s − 49-s + 1.37·53-s − 0.781·59-s + 0.768·61-s − 1.46·67-s − 0.237·71-s − 0.702·73-s − 1.80·79-s − 0.658·83-s − 0.423·89-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

\(L(1)\) \(\approx\) \(2.698070662\)
\(L(\frac12)\) \(\approx\) \(2.698070662\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
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

−13.15630669548725, −12.55111060980002, −12.20099070848266, −11.70578938634921, −11.31704594829010, −10.84955285956033, −10.28855605830723, −9.701088500319142, −9.334070618446349, −8.818679260625095, −8.611834621987324, −7.722642298115513, −7.194651037875845, −7.040447519227399, −6.335865287623956, −5.695158588555209, −5.520742775690209, −4.595710646742709, −4.193173129251858, −3.693893784860427, −3.140176340853963, −2.494926474300258, −1.635409117865603, −1.321412572376089, −0.4819229395445539, 0.4819229395445539, 1.321412572376089, 1.635409117865603, 2.494926474300258, 3.140176340853963, 3.693893784860427, 4.193173129251858, 4.595710646742709, 5.520742775690209, 5.695158588555209, 6.335865287623956, 7.040447519227399, 7.194651037875845, 7.722642298115513, 8.611834621987324, 8.818679260625095, 9.334070618446349, 9.701088500319142, 10.28855605830723, 10.84955285956033, 11.31704594829010, 11.70578938634921, 12.20099070848266, 12.55111060980002, 13.15630669548725

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