Properties

Label 2-33600-1.1-c1-0-23
Degree $2$
Conductor $33600$
Sign $1$
Analytic cond. $268.297$
Root an. cond. $16.3797$
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 − 6·11-s − 13-s − 3·17-s + 4·19-s − 21-s + 3·23-s + 27-s − 3·29-s + 5·31-s − 6·33-s − 10·37-s − 39-s + 9·41-s − 43-s + 49-s − 3·51-s + 9·53-s + 4·57-s − 9·59-s − 11·61-s − 63-s − 4·67-s + 3·69-s − 12·71-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s − 1.80·11-s − 0.277·13-s − 0.727·17-s + 0.917·19-s − 0.218·21-s + 0.625·23-s + 0.192·27-s − 0.557·29-s + 0.898·31-s − 1.04·33-s − 1.64·37-s − 0.160·39-s + 1.40·41-s − 0.152·43-s + 1/7·49-s − 0.420·51-s + 1.23·53-s + 0.529·57-s − 1.17·59-s − 1.40·61-s − 0.125·63-s − 0.488·67-s + 0.361·69-s − 1.42·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(33600\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(268.297\)
Root analytic conductor: \(16.3797\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{33600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 33600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.563889734\)
\(L(\frac12)\) \(\approx\) \(1.563889734\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−15.14985854721855, −14.47891884430368, −13.72509812428560, −13.49814108774471, −13.07135352645365, −12.33417159551918, −12.08167936439818, −11.01030262223176, −10.82909869971930, −10.09984109636650, −9.713558977618770, −9.004120530387878, −8.588242728997752, −7.873365589909299, −7.389250412196685, −7.026971557341628, −6.112017865609302, −5.525509984256517, −4.909347881388996, −4.377431754321866, −3.445459026870211, −2.874478517836565, −2.449923557027374, −1.556970845026160, −0.4390385415804812, 0.4390385415804812, 1.556970845026160, 2.449923557027374, 2.874478517836565, 3.445459026870211, 4.377431754321866, 4.909347881388996, 5.525509984256517, 6.112017865609302, 7.026971557341628, 7.389250412196685, 7.873365589909299, 8.588242728997752, 9.004120530387878, 9.713558977618770, 10.09984109636650, 10.82909869971930, 11.01030262223176, 12.08167936439818, 12.33417159551918, 13.07135352645365, 13.49814108774471, 13.72509812428560, 14.47891884430368, 15.14985854721855

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