Properties

Label 2-32340-1.1-c1-0-10
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 − 15-s + 2·17-s + 6·19-s + 8·23-s + 25-s − 27-s − 4·31-s + 33-s − 2·37-s − 8·41-s + 8·43-s + 45-s + 12·47-s − 2·51-s − 10·53-s − 55-s − 6·57-s + 12·59-s + 2·61-s − 4·67-s − 8·69-s − 8·71-s + 12·73-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.301·11-s − 0.258·15-s + 0.485·17-s + 1.37·19-s + 1.66·23-s + 1/5·25-s − 0.192·27-s − 0.718·31-s + 0.174·33-s − 0.328·37-s − 1.24·41-s + 1.21·43-s + 0.149·45-s + 1.75·47-s − 0.280·51-s − 1.37·53-s − 0.134·55-s − 0.794·57-s + 1.56·59-s + 0.256·61-s − 0.488·67-s − 0.963·69-s − 0.949·71-s + 1.40·73-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

\(L(1)\) \(\approx\) \(2.346070474\)
\(L(\frac12)\) \(\approx\) \(2.346070474\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−15.06179161384163, −14.42444841314774, −14.07118421602284, −13.23057810187980, −13.11930929825416, −12.29760098507285, −11.95407279241655, −11.23000964697779, −10.85789421855554, −10.23226244854298, −9.767232642924492, −9.106603007195371, −8.744887373592634, −7.774873507287412, −7.329468684810624, −6.863834698593887, −6.098871521444175, −5.437950093653248, −5.204108575132612, −4.482914763864828, −3.552609617529418, −3.040442563069127, −2.196081598006722, −1.291657716718942, −0.6641632439699440, 0.6641632439699440, 1.291657716718942, 2.196081598006722, 3.040442563069127, 3.552609617529418, 4.482914763864828, 5.204108575132612, 5.437950093653248, 6.098871521444175, 6.863834698593887, 7.329468684810624, 7.774873507287412, 8.744887373592634, 9.106603007195371, 9.767232642924492, 10.23226244854298, 10.85789421855554, 11.23000964697779, 11.95407279241655, 12.29760098507285, 13.11930929825416, 13.23057810187980, 14.07118421602284, 14.42444841314774, 15.06179161384163

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