Properties

Label 2-32340-1.1-c1-0-24
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s + 9-s − 11-s + 13-s − 15-s − 4·17-s − 6·19-s − 23-s + 25-s − 27-s + 5·29-s + 33-s − 39-s − 3·41-s + 43-s + 45-s − 7·47-s + 4·51-s − 3·53-s − 55-s + 6·57-s − 8·59-s + 6·61-s + 65-s + 14·67-s + 69-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.301·11-s + 0.277·13-s − 0.258·15-s − 0.970·17-s − 1.37·19-s − 0.208·23-s + 1/5·25-s − 0.192·27-s + 0.928·29-s + 0.174·33-s − 0.160·39-s − 0.468·41-s + 0.152·43-s + 0.149·45-s − 1.02·47-s + 0.560·51-s − 0.412·53-s − 0.134·55-s + 0.794·57-s − 1.04·59-s + 0.768·61-s + 0.124·65-s + 1.71·67-s + 0.120·69-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: \(1\)
Selberg data: \((2,\ 32340,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
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

−15.41267650276420, −14.81529320033364, −14.23732898350052, −13.60530015488128, −13.21926576048631, −12.62402474950037, −12.26627674547459, −11.49577236413436, −10.94828502026113, −10.66192512814796, −10.00289100494079, −9.489742540476247, −8.802095557684405, −8.288824940596881, −7.756123075628178, −6.770640220733496, −6.497585364645720, −6.072681951434813, −5.122343767883495, −4.840037384005664, −4.068155787857129, −3.395956576289063, −2.362485671958040, −1.980997385034965, −0.9301383370382955, 0, 0.9301383370382955, 1.980997385034965, 2.362485671958040, 3.395956576289063, 4.068155787857129, 4.840037384005664, 5.122343767883495, 6.072681951434813, 6.497585364645720, 6.770640220733496, 7.756123075628178, 8.288824940596881, 8.802095557684405, 9.489742540476247, 10.00289100494079, 10.66192512814796, 10.94828502026113, 11.49577236413436, 12.26627674547459, 12.62402474950037, 13.21926576048631, 13.60530015488128, 14.23732898350052, 14.81529320033364, 15.41267650276420

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