Properties

Label 2-369600-1.1-c1-0-111
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 − 5·13-s + 3·17-s + 3·19-s − 21-s − 4·23-s + 27-s + 10·31-s + 33-s + 37-s − 5·39-s − 7·41-s + 10·43-s − 10·47-s + 49-s + 3·51-s + 53-s + 3·57-s + 6·59-s − 61-s − 63-s − 5·67-s − 4·69-s − 9·71-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s − 1.38·13-s + 0.727·17-s + 0.688·19-s − 0.218·21-s − 0.834·23-s + 0.192·27-s + 1.79·31-s + 0.174·33-s + 0.164·37-s − 0.800·39-s − 1.09·41-s + 1.52·43-s − 1.45·47-s + 1/7·49-s + 0.420·51-s + 0.137·53-s + 0.397·57-s + 0.781·59-s − 0.128·61-s − 0.125·63-s − 0.610·67-s − 0.481·69-s − 1.06·71-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

\(L(1)\) \(\approx\) \(2.597532839\)
\(L(\frac12)\) \(\approx\) \(2.597532839\)
\(L(\frac{3}{2})\) 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 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 3 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−12.44585034934942, −12.02667298315235, −11.78008474258137, −11.27929614766359, −10.42414588598834, −10.12077512904793, −9.829455134941953, −9.425210904210859, −8.898416441107478, −8.364137781223687, −7.865549422556854, −7.531259855106902, −7.099299732716836, −6.382363838823954, −6.230667512406352, −5.262050807919402, −5.155433359446327, −4.336066296127704, −4.037960063712602, −3.289185304195719, −2.884966272373617, −2.453312748184026, −1.765075552548590, −1.131491372261151, −0.4152647462078905, 0.4152647462078905, 1.131491372261151, 1.765075552548590, 2.453312748184026, 2.884966272373617, 3.289185304195719, 4.037960063712602, 4.336066296127704, 5.155433359446327, 5.262050807919402, 6.230667512406352, 6.382363838823954, 7.099299732716836, 7.531259855106902, 7.865549422556854, 8.364137781223687, 8.898416441107478, 9.425210904210859, 9.829455134941953, 10.12077512904793, 10.42414588598834, 11.27929614766359, 11.78008474258137, 12.02667298315235, 12.44585034934942

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