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Properties

Label 2-32340-1.1-c1-0-20
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 + 6·13-s + 15-s − 2·17-s + 6·23-s + 25-s + 27-s + 6·29-s + 2·31-s + 33-s + 10·37-s + 6·39-s + 8·41-s − 8·43-s + 45-s − 4·47-s − 2·51-s + 6·53-s + 55-s + 6·59-s + 8·61-s + 6·65-s + 14·67-s + 6·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.301·11-s + 1.66·13-s + 0.258·15-s − 0.485·17-s + 1.25·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.359·31-s + 0.174·33-s + 1.64·37-s + 0.960·39-s + 1.24·41-s − 1.21·43-s + 0.149·45-s − 0.583·47-s − 0.280·51-s + 0.824·53-s + 0.134·55-s + 0.781·59-s + 1.02·61-s + 0.744·65-s + 1.71·67-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

\(L(1)\) \(\approx\) \(4.543101607\)
\(L(\frac12)\) \(\approx\) \(4.543101607\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−14.89101800201261, −14.54047199237531, −14.00471302732117, −13.28558453693965, −13.18370755442093, −12.67087179993497, −11.66965728976239, −11.37236259059953, −10.78613409320356, −10.16500838986247, −9.641488352591159, −9.016267382231998, −8.551779706792736, −8.215510574202502, −7.372762247447798, −6.670275254824785, −6.348143611214265, −5.651603895590448, −4.903870279428822, −4.219262137287102, −3.665150886193158, −2.887388123851232, −2.366304853246032, −1.340834922397187, −0.8886722703146049, 0.8886722703146049, 1.340834922397187, 2.366304853246032, 2.887388123851232, 3.665150886193158, 4.219262137287102, 4.903870279428822, 5.651603895590448, 6.348143611214265, 6.670275254824785, 7.372762247447798, 8.215510574202502, 8.551779706792736, 9.016267382231998, 9.641488352591159, 10.16500838986247, 10.78613409320356, 11.37236259059953, 11.66965728976239, 12.67087179993497, 13.18370755442093, 13.28558453693965, 14.00471302732117, 14.54047199237531, 14.89101800201261

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