Properties

Label 2-369600-1.1-c1-0-116
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 − 6·13-s + 6·17-s + 2·19-s + 21-s − 27-s − 8·29-s + 33-s + 10·37-s + 6·39-s + 8·41-s − 6·47-s + 49-s − 6·51-s − 2·53-s − 2·57-s − 8·59-s + 14·61-s − 63-s − 14·67-s − 8·71-s + 10·73-s + 77-s + 14·79-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.66·13-s + 1.45·17-s + 0.458·19-s + 0.218·21-s − 0.192·27-s − 1.48·29-s + 0.174·33-s + 1.64·37-s + 0.960·39-s + 1.24·41-s − 0.875·47-s + 1/7·49-s − 0.840·51-s − 0.274·53-s − 0.264·57-s − 1.04·59-s + 1.79·61-s − 0.125·63-s − 1.71·67-s − 0.949·71-s + 1.17·73-s + 0.113·77-s + 1.57·79-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

\(L(1)\) \(\approx\) \(1.626674264\)
\(L(\frac12)\) \(\approx\) \(1.626674264\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−12.44506440807369, −12.06962305309919, −11.65410847329011, −11.20013989591070, −10.71943511032922, −10.09512640243385, −9.867423147073493, −9.421872396609724, −9.132795556998821, −8.237615832795162, −7.706195410831208, −7.454526861960376, −7.187229910536928, −6.299267307539577, −6.022927321853355, −5.488734337333729, −5.039742236903528, −4.633684213372473, −4.004661667341116, −3.374288261036591, −2.928827110655332, −2.303965530761623, −1.748984882863145, −0.8866091231586141, −0.4202370213291917, 0.4202370213291917, 0.8866091231586141, 1.748984882863145, 2.303965530761623, 2.928827110655332, 3.374288261036591, 4.004661667341116, 4.633684213372473, 5.039742236903528, 5.488734337333729, 6.022927321853355, 6.299267307539577, 7.187229910536928, 7.454526861960376, 7.706195410831208, 8.237615832795162, 9.132795556998821, 9.421872396609724, 9.867423147073493, 10.09512640243385, 10.71943511032922, 11.20013989591070, 11.65410847329011, 12.06962305309919, 12.44506440807369

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