Properties

Label 2-32340-1.1-c1-0-2
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 − 13-s − 15-s − 6·17-s − 2·19-s − 6·23-s + 25-s + 27-s − 8·29-s − 31-s − 33-s − 6·37-s − 39-s + 11·43-s − 45-s + 10·47-s − 6·51-s − 2·53-s + 55-s − 2·57-s − 5·59-s + 2·61-s + 65-s + 2·67-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.301·11-s − 0.277·13-s − 0.258·15-s − 1.45·17-s − 0.458·19-s − 1.25·23-s + 1/5·25-s + 0.192·27-s − 1.48·29-s − 0.179·31-s − 0.174·33-s − 0.986·37-s − 0.160·39-s + 1.67·43-s − 0.149·45-s + 1.45·47-s − 0.840·51-s − 0.274·53-s + 0.134·55-s − 0.264·57-s − 0.650·59-s + 0.256·61-s + 0.124·65-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

\(L(1)\) \(\approx\) \(1.130584373\)
\(L(\frac12)\) \(\approx\) \(1.130584373\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 7 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
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

−15.14079459064141, −14.56413401181122, −13.91403274298874, −13.58938929370527, −12.90197539661687, −12.48897369839749, −11.95116243256075, −11.23877660885515, −10.77238715650823, −10.32385576570422, −9.528783818172347, −9.038646080728385, −8.635972924888547, −7.910828072163000, −7.472644546894797, −6.961795863239784, −6.211039007949465, −5.622488179092698, −4.837121690693997, −4.053155807900808, −3.917838132951167, −2.866731759631982, −2.263231105458892, −1.679048122727143, −0.3669113760550553, 0.3669113760550553, 1.679048122727143, 2.263231105458892, 2.866731759631982, 3.917838132951167, 4.053155807900808, 4.837121690693997, 5.622488179092698, 6.211039007949465, 6.961795863239784, 7.472644546894797, 7.910828072163000, 8.635972924888547, 9.038646080728385, 9.528783818172347, 10.32385576570422, 10.77238715650823, 11.23877660885515, 11.95116243256075, 12.48897369839749, 12.90197539661687, 13.58938929370527, 13.91403274298874, 14.56413401181122, 15.14079459064141

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