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Properties

Label 2-32340-1.1-c1-0-16
Degree $2$
Conductor $32340$
Sign $1$
Analytic cond. $258.236$
Root an. cond. $16.0697$
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 + 5-s + 9-s − 11-s + 5·13-s + 15-s + 6·17-s + 2·19-s − 6·23-s + 25-s + 27-s − 7·31-s − 33-s + 2·37-s + 5·39-s − 43-s + 45-s + 6·47-s + 6·51-s + 6·53-s − 55-s + 2·57-s − 3·59-s − 10·61-s + 5·65-s + 2·67-s − 6·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.301·11-s + 1.38·13-s + 0.258·15-s + 1.45·17-s + 0.458·19-s − 1.25·23-s + 1/5·25-s + 0.192·27-s − 1.25·31-s − 0.174·33-s + 0.328·37-s + 0.800·39-s − 0.152·43-s + 0.149·45-s + 0.875·47-s + 0.840·51-s + 0.824·53-s − 0.134·55-s + 0.264·57-s − 0.390·59-s − 1.28·61-s + 0.620·65-s + 0.244·67-s − 0.722·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32340\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(258.236\)
Root analytic conductor: \(16.0697\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 32340,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.883299073\)
\(L(\frac12)\) \(\approx\) \(3.883299073\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 \)
11 \( 1 + T \)
good13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 15 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
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

−15.15326527777205, −14.23399669690853, −14.08913820246909, −13.57820735111552, −13.05538184282984, −12.36252770218651, −12.07719733561076, −11.19955304265592, −10.71563937684154, −10.23232677012070, −9.515715407890319, −9.278677370750123, −8.465604480772796, −7.976252260226982, −7.595350042590410, −6.784205989570042, −6.133862990939224, −5.592649996325333, −5.132070130118811, −4.049485481560454, −3.667362586958516, −3.038783441109188, −2.192289913441627, −1.525836613332961, −0.7528827367910720, 0.7528827367910720, 1.525836613332961, 2.192289913441627, 3.038783441109188, 3.667362586958516, 4.049485481560454, 5.132070130118811, 5.592649996325333, 6.133862990939224, 6.784205989570042, 7.595350042590410, 7.976252260226982, 8.465604480772796, 9.278677370750123, 9.515715407890319, 10.23232677012070, 10.71563937684154, 11.19955304265592, 12.07719733561076, 12.36252770218651, 13.05538184282984, 13.57820735111552, 14.08913820246909, 14.23399669690853, 15.15326527777205

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