Properties

Label 2-369600-1.1-c1-0-129
Degree $2$
Conductor $369600$
Sign $1$
Analytic cond. $2951.27$
Root an. cond. $54.3256$
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  + 3-s − 7-s + 9-s + 11-s − 2·13-s + 6·17-s − 4·19-s − 21-s + 8·23-s + 27-s − 6·29-s − 8·31-s + 33-s + 6·37-s − 2·39-s − 6·41-s + 4·43-s − 8·47-s + 49-s + 6·51-s − 10·53-s − 4·57-s − 12·59-s + 10·61-s − 63-s + 12·67-s + 8·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.554·13-s + 1.45·17-s − 0.917·19-s − 0.218·21-s + 1.66·23-s + 0.192·27-s − 1.11·29-s − 1.43·31-s + 0.174·33-s + 0.986·37-s − 0.320·39-s − 0.937·41-s + 0.609·43-s − 1.16·47-s + 1/7·49-s + 0.840·51-s − 1.37·53-s − 0.529·57-s − 1.56·59-s + 1.28·61-s − 0.125·63-s + 1.46·67-s + 0.963·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(369600\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(2951.27\)
Root analytic conductor: \(54.3256\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{369600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 369600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.993434079\)
\(L(\frac12)\) \(\approx\) \(2.993434079\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 \)
7 \( 1 + T \)
11 \( 1 - T \)
good13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
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 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−12.62072134843622, −12.22962380179754, −11.50801815362473, −11.09745702452866, −10.80392195589868, −10.05752108002196, −9.723941619453783, −9.373974404756433, −8.931150143594728, −8.424999295748899, −7.899831654375842, −7.361278261793114, −7.215661688256847, −6.410778950495960, −6.165292076429043, −5.370937735917938, −5.033319072077900, −4.516243935400692, −3.744628860608098, −3.395458705290793, −3.067636606043924, −2.226357731044143, −1.862775799079023, −1.110673127553509, −0.4507994496043278, 0.4507994496043278, 1.110673127553509, 1.862775799079023, 2.226357731044143, 3.067636606043924, 3.395458705290793, 3.744628860608098, 4.516243935400692, 5.033319072077900, 5.370937735917938, 6.165292076429043, 6.410778950495960, 7.215661688256847, 7.361278261793114, 7.899831654375842, 8.424999295748899, 8.931150143594728, 9.373974404756433, 9.723941619453783, 10.05752108002196, 10.80392195589868, 11.09745702452866, 11.50801815362473, 12.22962380179754, 12.62072134843622

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