Properties

Label 2-32340-1.1-c1-0-9
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 − 2·17-s + 19-s + 2·23-s + 25-s + 27-s + 6·29-s + 31-s − 33-s + 37-s + 5·39-s − 8·41-s + 43-s − 45-s − 2·47-s − 2·51-s − 4·53-s + 55-s + 57-s − 6·59-s + 14·61-s − 5·65-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 − 0.485·17-s + 0.229·19-s + 0.417·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.179·31-s − 0.174·33-s + 0.164·37-s + 0.800·39-s − 1.24·41-s + 0.152·43-s − 0.149·45-s − 0.291·47-s − 0.280·51-s − 0.549·53-s + 0.134·55-s + 0.132·57-s − 0.781·59-s + 1.79·61-s − 0.620·65-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

\(L(1)\) \(\approx\) \(2.842318540\)
\(L(\frac12)\) \(\approx\) \(2.842318540\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 13 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−15.12418428626659, −14.49813230733012, −13.96030717391721, −13.43414559750224, −13.11578142660400, −12.43454774163194, −11.88408330823099, −11.22351435950809, −10.88435857082073, −10.19072568977705, −9.706668170547266, −8.873141679434614, −8.583846056317321, −8.094193906194863, −7.481150679358722, −6.765301344979548, −6.362016673226565, −5.579973199494340, −4.808898021593203, −4.309678406891781, −3.489111987566853, −3.149545945008321, −2.283544690607354, −1.452774389342246, −0.6450663244224321, 0.6450663244224321, 1.452774389342246, 2.283544690607354, 3.149545945008321, 3.489111987566853, 4.309678406891781, 4.808898021593203, 5.579973199494340, 6.362016673226565, 6.765301344979548, 7.481150679358722, 8.094193906194863, 8.583846056317321, 8.873141679434614, 9.706668170547266, 10.19072568977705, 10.88435857082073, 11.22351435950809, 11.88408330823099, 12.43454774163194, 13.11578142660400, 13.43414559750224, 13.96030717391721, 14.49813230733012, 15.12418428626659

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