Properties

Label 2-32340-1.1-c1-0-5
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 − 2·13-s − 15-s + 6·17-s − 8·19-s − 6·23-s + 25-s − 27-s + 6·29-s − 2·31-s + 33-s + 2·37-s + 2·39-s + 8·43-s + 45-s + 12·47-s − 6·51-s + 6·53-s − 55-s + 8·57-s − 6·59-s − 8·61-s − 2·65-s + 2·67-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.301·11-s − 0.554·13-s − 0.258·15-s + 1.45·17-s − 1.83·19-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 + 0.328·37-s + 0.320·39-s + 1.21·43-s + 0.149·45-s + 1.75·47-s − 0.840·51-s + 0.824·53-s − 0.134·55-s + 1.05·57-s − 0.781·59-s − 1.02·61-s − 0.248·65-s + 0.244·67-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: $\chi_{32340} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 32340,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.505849866\)
\(L(\frac12)\) \(\approx\) \(1.505849866\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−14.99463205666567, −14.54613525406964, −13.91845656715472, −13.60179799064420, −12.65669287619881, −12.36577154681133, −12.15045560860605, −11.25136670407632, −10.60194709490337, −10.38257279673239, −9.787277963334880, −9.229532210168243, −8.492360064735371, −7.928781520160039, −7.407684784160954, −6.685381622147784, −6.074849845628143, −5.698095729903304, −5.059306199179439, −4.296471182346595, −3.879476625065096, −2.769178037637018, −2.283996813676282, −1.407219077667989, −0.4909082592640266, 0.4909082592640266, 1.407219077667989, 2.283996813676282, 2.769178037637018, 3.879476625065096, 4.296471182346595, 5.059306199179439, 5.698095729903304, 6.074849845628143, 6.685381622147784, 7.407684784160954, 7.928781520160039, 8.492360064735371, 9.229532210168243, 9.787277963334880, 10.38257279673239, 10.60194709490337, 11.25136670407632, 12.15045560860605, 12.36577154681133, 12.65669287619881, 13.60179799064420, 13.91845656715472, 14.54613525406964, 14.99463205666567

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