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Properties

Label 2-32340-1.1-c1-0-6
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 + 3·17-s − 3·19-s − 23-s + 25-s − 27-s − 7·29-s − 4·31-s − 33-s − 6·37-s − 6·39-s + 4·41-s − 9·43-s − 45-s − 3·51-s + 3·53-s − 55-s + 3·57-s − 5·59-s − 5·61-s − 6·65-s − 2·67-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 + 0.727·17-s − 0.688·19-s − 0.208·23-s + 1/5·25-s − 0.192·27-s − 1.29·29-s − 0.718·31-s − 0.174·33-s − 0.986·37-s − 0.960·39-s + 0.624·41-s − 1.37·43-s − 0.149·45-s − 0.420·51-s + 0.412·53-s − 0.134·55-s + 0.397·57-s − 0.650·59-s − 0.640·61-s − 0.744·65-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 32340,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.445976630\)
\(L(\frac12)\) \(\approx\) \(1.445976630\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + 9 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−15.03661525274663, −14.65177982180317, −13.94654088259004, −13.31966689222602, −13.03221966292837, −12.16004038662976, −12.02860734360576, −11.15059007878006, −10.93367756725930, −10.44544456689862, −9.658759667942841, −9.095835173978363, −8.528272394692121, −7.981245068414760, −7.372337389137991, −6.711114282119059, −6.171189992856228, −5.658888156556156, −5.038749763362399, −4.226208956650280, −3.652407571690197, −3.283715765943966, −2.000506591231335, −1.418706067053713, −0.4938637652996152, 0.4938637652996152, 1.418706067053713, 2.000506591231335, 3.283715765943966, 3.652407571690197, 4.226208956650280, 5.038749763362399, 5.658888156556156, 6.171189992856228, 6.711114282119059, 7.372337389137991, 7.981245068414760, 8.528272394692121, 9.095835173978363, 9.658759667942841, 10.44544456689862, 10.93367756725930, 11.15059007878006, 12.02860734360576, 12.16004038662976, 13.03221966292837, 13.31966689222602, 13.94654088259004, 14.65177982180317, 15.03661525274663

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