Properties

Label 2-32340-1.1-c1-0-31
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s + 9-s + 11-s − 2·13-s − 15-s + 2·17-s − 6·19-s + 25-s + 27-s + 2·29-s + 4·31-s + 33-s − 2·37-s − 2·39-s + 4·43-s − 45-s + 8·47-s + 2·51-s − 4·53-s − 55-s − 6·57-s − 14·59-s + 14·61-s + 2·65-s − 2·67-s − 8·71-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 + 0.485·17-s − 1.37·19-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.718·31-s + 0.174·33-s − 0.328·37-s − 0.320·39-s + 0.609·43-s − 0.149·45-s + 1.16·47-s + 0.280·51-s − 0.549·53-s − 0.134·55-s − 0.794·57-s − 1.82·59-s + 1.79·61-s + 0.248·65-s − 0.244·67-s − 0.949·71-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: \(1\)
Selberg data: \((2,\ 32340,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−15.08303901075011, −14.89502000037363, −14.26757440585383, −13.87216608307921, −13.15810947069469, −12.67017260854547, −12.16042358669868, −11.74390770087871, −10.98001700920714, −10.44918441544131, −10.00106121560388, −9.305785824103527, −8.754914683284162, −8.339960283571038, −7.664276129631199, −7.239248347079112, −6.545387966419879, −5.994295527113702, −5.193505361021883, −4.369588694977418, −4.158846940247000, −3.232092534450764, −2.683137050231691, −1.938465028631644, −1.060275431478377, 0, 1.060275431478377, 1.938465028631644, 2.683137050231691, 3.232092534450764, 4.158846940247000, 4.369588694977418, 5.193505361021883, 5.994295527113702, 6.545387966419879, 7.239248347079112, 7.664276129631199, 8.339960283571038, 8.754914683284162, 9.305785824103527, 10.00106121560388, 10.44918441544131, 10.98001700920714, 11.74390770087871, 12.16042358669868, 12.67017260854547, 13.15810947069469, 13.87216608307921, 14.26757440585383, 14.89502000037363, 15.08303901075011

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